three essays on multivarite volatility modelling and...
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Three Essays on Multivariate Volatility Modelling and Estimation
Mustafa Hakan Eratalay
Three Essays on Multivariate VolatilityModelling and Estimation
by MUSTAFA HAKAN ERATALAY
Doctoral Dissertation
Supervisor: Prof. M. ANGELES CARNERO FERNANDEZ
Quantitative Economics DoctorateDepartamento de Fundamentos del Análisis Económico
Universidad de Alicante
June 2012
Three Essays on Multivariate VolatilityModelling and Estimation
by MUSTAFA HAKAN ERATALAY
Doctoral Dissertation
Supervisor: Prof. M. ANGELES CARNERO FERNANDEZ
Quantitative Economics DoctorateDepartamento de Fundamentos del Análisis Económico
Universidad de Alicante
June 2012
To my family...
Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1 Estimating VAR-MGARCH Models in Multiple Steps 311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2 Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3 Estimation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Vector Autoregressive CCC, ECCC, DCC and cDCC GARCH
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.3.2 Vector Autoregressive RSDC-GARCH model . . . . . . . . . . . . 42
1.4 Monte Carlo Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.4.1 Innovations Distributed as a Gaussian or Student-t . . . . . . . . 45
1.4.2 Robustness to Error Distribution . . . . . . . . . . . . . . . . . . 48
1.4.3 Robustness to Model . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4.4 Innovations Distributed as a Skewed Student-t . . . . . . . . . . . 51
1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 Estimation of Multivariate Stochastic Volatility Models: A Compara-tive Monte Carlo Study 772.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2 Multivariate Stochastic Volatility (MSV) Models . . . . . . . . . . . . . . 81
2.2.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.2.2 Time Varying Correlation MSV . . . . . . . . . . . . . . . . . . . 82
2.2.3 MSV with Leverage E¤ect . . . . . . . . . . . . . . . . . . . . . . 84
2.2.4 Estimating the MSV Models . . . . . . . . . . . . . . . . . . . . . 86
2.3 Monte Carlo Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4 An Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
1
2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3 Do Correlated Markets Have More Volatility Spillovers? 1213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.2 Econometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.4 Monte Carlo Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.4.1 Performance of the ML Estimator . . . . . . . . . . . . . . . . . . 127
3.4.2 VaR Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.4.3 Robustness to Model . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
2
Acknowledgements
First and foremost, I would like to thank Prof. Angeles Carnero who shared with me her
great knowledge and expertise. It is di¢ cult to overstate my appreciation of her helpful
suggestions and comments, constructive critics, and her support and motivation when
writing my thesis.
I would like to thank also Prof. Esther Ruiz (from Universidad Carlos III de Madrid),
and Prof. Siem Jan Koopman (from V.U. University Amsterdam) for introducing me
to the misterious world of stochastic volatility and for their directive suggestions in
developing the second chapter of my thesis, Prof. Marco van der Leij (from University of
Amsterdam) for his comments and critics on the third chapter of my thesis, Prof. Angel
León for his suggestions which helped me to improve the quality of the �rst chapter
of my thesis, and Prof. Juan Mora for being always available whenever I needed some
advice.
I am sincerely thankful to the members of the Deparment of Fundamentos del Análisis
Económico at the Universidad de Alicante, for being such wonderful people that I will
always remember. I also appreciate the help of Marilo Rufete and the secretarial sta¤
of the department about the administrative issues during my graduate studies. I would
like to thank my colleagues Rocio Alvarez, Carlos Aller, Lorenzo Ductor, Álex Perez,
Gergely Horváth, Jonas Hedlund and Gustavo Cabrera for �lling my PhD life with nice
memories.
I am grateful to my uncle Prof. Kenan Eratalay for suggesting me to build an
academic career. And �nally, this thesis would not have been possible without the
unconditional love and support of my parents Nevzat and Aynur Eratalay, my brothers
Süleyman and Selim Eratalay, and my better half Riia Arukaevu.
3
Resumen
Muchas series temporales �nancieras, como los retornos de los activos o los tipos de
cambio, muestran regularidades comúnes como la volatilidad que varía con el tiempo o
comovimientos. Estas regularidades se conocen en la literatura como hechos establecidos
(stylized facts). Un modelo deseable de series temporales es el que explica una o más de
estas regularidades. Algunos de estos hechos establecidos son los siguientes:
� La volatilidad agrupada (volatility clustering)
Se observa que períodos de alta (baja) volatilidad son seguidos por períodos de
baja (alta) volatilidad. Este tipo de comportamiento persistente sugiere que podría
haber una estructura autorregresiva que rige la dinámica de las volatilidades. Los
modelos ARCH introducidos por Engle (1982), los modelos SV introducidos por
Taylor (1986) y sus extensiones, a que se re�ere a lo largo de esta tesis, se han
desarrollado para simular esta propiedad de volatilidad agrupada.
� Las colas pesadas (thick tails)
Está documentado en Mandelbrot (1963) y Fama (1963, 1965) entre otros que
los retornos de los activos tienden a presentar una distribución de cola pesada
o leptocúrtica. Para coincidir con este hecho establecido, se utilizan diferentes
distribuciones en la literatura como la distribución de Student-t (véase, por ejemplo
Fiorentini et al. 2003, Sandmann y Koopman (1998)). El primer capítulo de esta
tesis trabajamos también con la distribución Student-t .
� Los efectos apalancamiento (leverage e¤ects)
El efecto palanca se re�ere a la correlación negativa entre los retornos y volatil-
idades: es decir, un retorno negativo se espera que aumente la volatilidad más
que un retorno positivo. La intuición es que una disminución en los precios de las
acciones implica un mayor apalancamiento de las empresas, lo que aumenta los
riesgos e incertidumbres, y por tanto aumenta la volatilidad. Se pueden encontrar
ejemplos en Nelson (1991) o Jungbacker y Koopman (2005). En el segundo capí-
tulo de esta tesis, proponemos dos modelos multivariantes para captar los efectos
de apalancamiento.
� Los comovimientos
Las transmisiones de volatilidad (volatility spillovers) y las correlaciones entre los
retornos han recibido cada vez más interés en la literatura. Entre otros Jeantheau
5
(1998), Longin y Solnik (1995), Bae y Karolyi (1994) analizan teórica y/o empíri-
camente los efectos de transmisión de volatilidad. Bollerslev (1990), Engle (2002)
de Mao Tse (2000), Pelletier (2006) son algunos de los trabajos que estudian las
correlaciones entre los retornos de los mercados de valores. El primer capítulo de
esta tesis considera el modelo de Jeantheau (1998) y el tercer capítulo propone
un modelo para captar las transmisiones de volatilidad. En todos los capítulos de
esta tesis los modelos de correlación constante y / o variable con el tiempo son
considerados.
Para explicar la volatilidad variable en el tiempo, Engle (1982) y Bollerslev (1996) pro-
pusieron modelos autoregresivos generalizados de heterocedasticidad condicional (GARCH).
En los modelos GARCH, las volatilidades siguen una función deterministica de los re-
tornos del día anterior al cuadrado y volatilidades. Por lo tanto las observaciones rigen
la dinámica de las volatilidades en estos modelos. Despues, los modelos GARCH se
han extendido a los modelos de GARCH multivariantes (MGARCH) para capturar los
efectos de transmisión de volatilidad y las correlación entre las series, véase, por ejem-
plo, Bauwens et al. (2006) y Silvennoinen y Teräsvirta (2009) para una encuesta de
los modelos multivariantes de GARCH. Entre otras, por ejemplo el modelo exponen-
cial GARCH (EGARCH) propuesto por Nelson (1991) se ha desarrollado para explicar
los efectos de apalancamiento, el modelo correlación condicional constante extendido
(ECCC) GARCH de Jeantheau (1998) ha sido desarrollado para capturar las transmi-
siones de volatilidad, Bollerslev (1990) propone el modelo de la correlación condicional
constante (CCC) GARCH para capturar la correlación entre las series y Engle (2002)
propone el modelo correlación condicional dinamico (DCC) GARCH para permitir que
estas correlaciones entre los retornos varíen con el tiempo.
Por otra parte, la literatura de la volatilidad estocástica (SV) iniciada por Taylor
(1986, 1994) y Hull y White (1987) sugiere modelizar la volatilidad variable en el tiempo
como un componente no observado y permite que su logaritmo siga un proceso autor-
regresivo. En este modelo, los parámetros rigen las volatilidades. Los modelos de SV
son atractivos en el sentido de que están más cerca de los modelos utilizados en la teoría
�nanciera para describir el comportamiento de los precios, véase Shephard y Andersen
(2008). Además, se ha demostrado que los modelos SV explican el comportamiento de las
volatilidades con más precisión en comparación con los modelos GARCH; véase Daniels-
son (1994), Kim et al. (1998) y Carnero et al. (2004). Aunque estadísticamente son más
atractivo que los modelos GARCH, los modelos SV tienen una desventaja en términos
de estimación, ya que las funciones de verosimilitud exacta de estos modelos son difíciles
de evaluar. Se han desarrollado varios modelos multivariantes de SV en la literatura.
Entre otros, Asai y McAleer (2006) propone el modelo MSV con apalancamiento para
6
explicar los efectos apalancamiento en un conjunto de datos de series temporales, Harvey
et al. (1994) propone el modelo CC-MSV para captar la correlación entre los retornos de
un conjunto de datos de series temporales, mientras que Jungbacker y Koopman (2006)
propone el modelo de MSV con correlaciones variables en el tiempo para permitir que
estas correlaciones entre retornos cambien con el tiempo.
El objetivo principal de la tesis es comparar los modelos GARCH y volatilidad es-
tocástica para explicar uno o más de los hechos establecidos en la literatura de series tem-
porales, y analizar el rendimiento de los métodos de estimación estimando los parámetros
de los modelos para las pequeñas muestras. En particular, uno de los objetivos de la
tesis es analizar el comportamiento de los estimadores de los parámetros de los modelos
GARCH multivariantes en múltiples etapas para muestras pequeñas en el caso de er-
rores Gaussianos y Student-t. Cuando estamos tratando de estimar un modelo GARCH
multivariante con muestras de gran tamaño o elevado número de series, la estimación
de los parámetros de la media, la varianza y correlación en diferentes etapas hace que el
proceso de estimación sea mucho más fácil. Por otro lado, perdemos de e�ciencia cuando
se estiman los parámetros en varios pasos. Por lo tanto la pregunta en este capítulo es
¿podríamos estimar los parámetros en varios pasos? ¿O deberíamos preferir estimar los
parámetros en una sola etapa? Otro objetivo de esta tesis es comparar los rendimien-
tos de la cuasi-máxima verosimilitud (QML) y Monte Carlo verosimilitud (MCL) en
la estimación de varios modelos multivariantes de volatilidad estocástica. El método
QML es relativamente fácil de aplicar y es más �exible para muestras grandes o altos
números de series, sin embargo, se basa en aproximaciones y por lo tanto es ine�ciente.
MCL es asintóticamente e�ciente, pero necesita mucho más tiempo para converger. Si
se considera muestras grandes o altos números de series, se enfrenta a di�cultades en la
aplicación de este método. Por lo tanto, tratamos de responder a la pregunta de si hay
algunos modelos o valores de los parámetros para los cuales el método QML se aproxima
al método MCL. Por otro lado, el efecto apalancamiento, como se ha de�nido anteri-
ormente, se re�ere a la correlación entre el retorno y la volatilidad futura de una serie.
En este trabajo tenemos la intención de desarrollar un modelo MSV con apalancamiento
que permite que los retornos de una serie se correlacionen con la volatilidad futura de la
otra serie. El tercer y último objetivo de la tesis es desarrollar un nuevo modelo GARCH
para capturar las transmisiones de volatilidad con menor número de parámetros. En la
literatura los modelos populares, como los modelos BEKK y ECCC-GARCH, requieren
una estimación de un elevado número de parámetros. Por ello, con estos modelos es
difícil estimar las transmisiones de volatilidad cuando se considera un gran número de
series. El modelo que proponemos en este capítulo utiliza las correlaciones variables en el
tiempo para capturar las transmisiones de volatilidad. Usando este modelo, nuestro ob-
7
jetivo es responder a la pregunta de si se podría usar la dinámica de las correlaciones para
explicar parcialmente las transmisiones de volatilidad, reduciendo entonces el número de
parámetros para estimar.
En todos los capítulos de la tesis, el plan de estudio consiste en explicar los an-
tecedentes teóricos y econométricos de los modelos y métodos de estimación y realizar
experimentos de simulación para analizar las propiedades de los modelos o métodos de
que se trate en pequeñas muestras. Los resultados de los experimentos de simulación
están presentados en forma de tablas que consiste en las medias, errores estándar y raíz
de error cuadrática media. También incluimos las �guras que presentan las estimaciones
de densidades kernel de las diferencias entre las volatilidades y las correlaciones y sus
valores verdaderos. Además, cada capítulo contiene una estimación empírica para ilus-
trar los resultados usando los datos de los mercados de valores. Toda la programación e
implementación de los modelos y métodos de estimación se llevan a cabo en MATLAB.
Para cumplir los objetivos, esta tesis consta de tres capítulos independientes. La
hipótesis del primer capítulo es que cuando los errores son Gaussianos, se puede estimar
los parámetros de media, varianza y correlación de los modelos de correlación condi-
cional en múltiples etapas y no se pierde mucha e�ciencia en las pequeñas muestras en
comparación con la estimación en una etapa. Cuando los errores siguen la distribución
Student-t, se espera que esta hipótesis sea rechazada. La hipótesis del segundo capí-
tulo es que aunque hay modelos y valores de los parámetros para los que la estimación
cuasi-máxima verosimilitud (QML) se aproxima a la estimación Monte Carlo verosimil-
itud (MCL). La otra hipótesis es que el modelo multivariante de volatilidad estocástica
con apalancamiento entre los retornos de una serie y la volatilidad futura de otra serie,
que proponemos en este capítulo, se estima satisfactoriamente por el método MCL. La
hipótesis del tercer capítulo es que el nuevo modelo GARCH que desarrollamos puede
capturar las transmisiones de volatilidad entre las series, con pocos parámetros y por lo
tanto podría ser preferible para cuando se considera un elevado número de series.
A continuación, se presenta un breve resumen de los tres capítulos:
Capítulo I: Estimating VAR-MGARCH Models in MultipleSteps
En este capítulo analizamos el rendimiento de la estimación en múltiples etapas de los
parámetros de modelos vector autorregresivos GARCH multivariantes (VAR-MGARCH)
para muestras pequeñas. Para ello, consideramos cinco modelos de correlación condi-
cional GARCH dadas por diferentes especi�caciones de la dinámica de la correlación:
correlación condicional constante (CCC) GARCH de Bollerslev (1990), correlación condi-
cional dinámica (DCC) GARCH de Engle (2002), DCC-GARCH consistente (cDCC-
8
GARCH) de Aielli (2008), correlación condicional constante extendida (ECCC) GARCH
de Jeantheau (1998) y correlación dinámica con cambio de regimen (RSDC) GARCH de
Pelletier (2006). Para la estimación de los modelos de DCC y cDCC-GARCH se utiliza
el método de focalización de la correlación (correlation targeting, véase Engle 2009) que
reduce el número de parámetros a estimar mediante la sustitución de la matriz de covar-
ianza de largo plazo por la matriz de covarianza muestral. Aielli (2008) muestra que la
estimación de modelos DCC-GARCH mediante la focalización de la correlación produce
estimadores inconsistentes y sugiere una versión corregida del modelo DCC-GARCH,
al que nos referiremos como el modelo cDCC-GARCH. Por otro lado, es común encon-
trar una estructura dinámica de la media condicional que puede ser modelizada con un
VARMA (véase, por ejemplo Veiga y McAleer 2008a, 2008b). Por simplicidad suponemos
un VAR (1) para la especi�cación de las medias condicionales.
Una forma de estimar los parámetros de todos estos modelos de VAR-MGARCH es
estimarlos en una etapa maximizando el log-verosimilitud completo. Suponiendo que
la distribución de los errores es correctamente especi�cado, los estimadores obtenidos
son de máxima verosimilitud (ML) y son consistentes y asintóticamente normales. Si
la verdadera distribución no es Gaussiana, pero la estimación se basa en errores de
Gauss, entonces los estimadores son de cuasi-máxima verosimilitud (QML) y también
son consistentes y asintóticamente normales. Véase, por ejemplo Bollerslev y Wooldridge
(1992). Si la estimación se basa en errores Student-t y la distribución verdadera es
simétrica, los estimadores QML serán consistentes y asintóticamente normales. Véase
Newey y Steigerwald (1997).
Una alternativa a la maximización completa de la verosimilitud es utilizar estimadores
en múltiples etapas. Bajo el supuesto de Gaussianidad, podemos estimar los parámetros
en dos etapas, véase, por ejemplo, Engle (2002). En la primera etapa se puede estimar los
parámetros de la media y de la varianza a la vez y condicionando en las estimaciones de
estos parámetros, en una segunda etapa se puede estimar los parámetros de correlación.
Como Engle y Sheppard (2001) sugiere para el modelo DCC-GARCH. Los estimadores
de dos etapas serán consistentes y asintóticamente normales, pero ine�cientes.
Por último, se podría estimar los parámetros de estos modelos en tres etapas: en
primer lugar se estiman los parámetros de la media. A continuación, condicionando
en la estimación de los parámetros de la media, podemos estimar los parámetros de la
varianza y luego, condicionando en las estimaciones de los parámetros de la media y de
la varianza, se estima los parámetros de correlación. Este método de estimación en tres
etapas se menciona en Bauwens et al. (2006) y puede ser visto como una estimación de
dos etapas para una serie con media cero. Por lo tanto, estimadores en tres etapas son
también consistentes y asintóticamente normal, pero ine�cientes.
9
Aunque no tiene antecedentes teóricas, también tenemos en cuenta la estimación
en múltiples etapas suponiendo una distribución Student-t, donde en todas las etapas
se basa la estimación en esta distribución. Bauwens y Laurent (2005) y Jondeau y
Rockinger (2005) también analizan los estimadores de múltiples etapas basadas en la
distribución Student-t, sin embargo, en estos articulos se estiman los parámetros de
varianza suponiendo errores normales, mientras que la estimación de los parámetros de
correlación se basa en Student-t.
En este capítulo se realizan varios experimentos de Monte Carlo para analizar y com-
parar los rendimientos de las estimaciones en una, dos y tres etapas de los cinco modelos
de corelacion condicional GARCH mencionados anteriormente suponiendo errores nor-
males o de Student-t.
Obtenemos que cuando la estimación se basa en la distribución normal, los rendimien-
tos de los estimadores de una etapa y de múltiples etapas son muy similares. Sin em-
bargo, cuando la estimación se basa en la distribución Student-t, para algunos modelos
las diferencias entre los estimadores podría ser relevante. Por lo tanto, la estimación en
múltiples etapas basada en la distribución Student-t no es una buena idea.
También veri�camos la robustez de nuestros resultados con respecto a una incorrecta
especi�cación de la distribución del error o el modelo. Nuestros resultados muestran que
si la distribución verdadera de los errores es una Student-t, pero la estimación se basa en
la distribución normal, las estimaciones de densidad de kernel de las diferencias entre las
estimaciones de la volatilidad y de la correlación, obtenidas a partir de estimadores de una
etapa y de múltiples etapas, y sus valores verdaderos son muy similares. Análogamente,
si la distribución verdadera de los errores es Gaussiana pero la estimación se basa en
la distribución Student-t, obtenemos los mismos resultados que cuando la verdadera y
la supuesta distribución son una Student-t. Cuando el modelo está incorrectamente
especi�cado y la estimación se basa en errores normales, obtenemos que, en promedio,
volatilidades y correlaciones están relativamente bien estimadas incluso cuando se utiliza
un modelo mal especi�cado. Las estimaciones en múltiples etapas de las volatilidades
y correlaciones distan de sus valores verdaderos menos de un 2% de lo que lo hace las
estimaciones en una etapa del modelo correctamente especi�cado.
Cuando los errores siguen una distribución Student-t sesgada, pero la estimación se
hace suponiendo una distribución normal, obtenemos que las estimaciones de la densidad
de kernel de la diferencia entre las estimaciones de una etapa y de múltiples etapas de las
volatilidades y correlaciones y sus verdaderos valores son muy similares. Sin embargo,
esto no es cierto cuando la estimación se basa en errores t de Student no sesgada. En
cualquier caso, cuando la verdadera distribución está sesgada, se debe ser prudente en el
uso de los estimadores de una etapa o múltiples etapas basados en los errores Student-t
10
ya que ambos son estimadores inconsistentes.
Capítulo II: Estimation of Multivariate Stochastic VolatilityModels: A Comparative Monte Carlo Study
En los modelos de volatilidad estocástica, las volatilidades variables en el tiempo son
consideradas como una componente no observada de tal manera que las log-volatilidades
se las deja a seguir un proceso autorregresivo? Como se mencionó anteriormente, aunque
los modelos de volatilidad estocástica son estadísticamente más atractivos, son más difí-
ciles en términos de estimación debido a que sus funciones de verosimilitud exactas son
difíciles de evaluar.
Harvey et al. (1994) propuso el método de la cuasi-máxima verosimilitud (QML)
para estimar los modelos SV multivariantes. Su método trabaja con la transformación
log-cuadrados de las observaciones. La distribución de los errores log-cuadrados en la
ecuación de observación se aproxima mediante una distribución normal, por lo tanto,
este método se basa en aproximaciones. Si tomamos las log-volatilidades como una
ecuación de estado, entonces obtenemos una ecuación de la forma de un estado de espacio
Gaussiano. Después de esto, se utiliza un �ltro de Kalman para construir la verosimilitud
que se maximiza para obtener los estimadores. Ruiz (1994) muestra que este método
QML es consistente y asintóticamente normal. La estimación QML es muy fácil de
aplicar, sin embargo, es ine�ciente dado que se basa en aproximaciones. Véase Jacquier
et al. (1994), Breidt y Carriquiry (1996) y Sandmann y Koopman (2005) como ejemplos.
La evaluación de la función de verosimilitud exacta requiere una integración de alta
dimensión que se podría obtener por métodos de simulación. Un método que logra esto
es el método de estimación Monte Carlo Verosimilitud (MCL) de Jungbacker y Koopman
(2006). Este método no utiliza una transformación log-cuadrado de las observaciones.
Como Durbin y Koopman (1997) mostró, el logaritmo de la verosimilitud de los modelos
de estado de espacio con los errores Gaussianos se puede escribir como una suma del
logaritmo de la verosimilitud del modelo de aproximación Gaussiano y una corrección
que corresponde a las desviaciones de la asunción Gaussiana con respecto al modelo real.
Lo que hace este método de máxima verosimilitud simulada más atractivo que otros, es
que las simulaciones sólo son necesarios para evaluar la parte de corrección en la log-
verosimilitud. Por lo tanto, este método requiere menos cantidad de simulaciones para
evaluar el logaritmo de la verosimilitud y toma mucho menos tiempo para hacerlo. El
método de MCL de Jungbacker y Koopman (2006) es consistente, asintóticamente nor-
mal y e�ciente. Cabe señalar que el método de MCL podría requerir transformaciones
especí�cas para estimar algunos modelos MSV, pero siempre y cuando estas transfor-
maciones se puedan hacer, el método MCL demuestra ser muy útil en la obtención de
11
estimadores asintóticamente e�cientes. También el método MCL requiere el cálculo de
los derivados con respecto a las variables de estado, que podrían obtenerse analíticamente
o numéricamente.
En este capítulo se comparan los rendimientos del método QML de Harvey et al.
(1994) y del método de MCL de Jungbacker y Koopman (2006) en la estimación de
los parámetros de los modelos MSV con correlación constante (CC) MSV de Harvey et
al. (1994), MSV con correlación variable en el tiempo (TVC) MSV de Jungbacker y
Koopman (2006) y de dos modelos MSV con el apalancamiento que proponemos para
las muestras pequeñas. Asai y McAleer (2006) también propuso un modelo MSV con
el apalancamiento, donde la matriz de apalancamiento es diagonal. Esto implica que
las innovaciones de volatilidad en el futuro de una serie sólo se correlaciona con las
innovaciones de retorno de esa misma serie. Por otra parte, estas correlaciones en su
artículo son proporcionales a las varianzas de las innovaciones de volatilidad, lo cual es
una restricción impuesta en el modelo. En esta tesis, proponemos un modelo MSV con el
apalancamiento diagonal, donde no hay ninguna restricción de este tipo en la correlación.
Además, para permitir la posible correlación entre los innovaciones de retorno de una
serie y las innovaciones de volatilidad de otra serie, proponemos el modelo MSV con el
apalancamiento no diagonal. Para la estimación de estos dos modelos con el método
de QML, adaptamos las transformaciones dadas en Asai y McAleer (2006). Para la
estimación con el método de MCL, ofrecemos las transformaciones necesarias.
Basándonos en los resultados, concluimos que a pesar de que en el caso de TVCMSV,
el rendimiento del estimador QML se aproxima al del estimador MCL, el segundo siem-
pre es preferible. No recomendamos el uso de los estimadores QML para los modelos
con apalancamiento. Aunque el método QML se puede implementar mucho más fácil-
mente que el MCL y el tiempo de cálculo es mucho menor en la estimación de QML,
sugerimos su uso si se espera que las series tengan una correlación alta y / o que cambie
con el tiempo y los procesos de SV tengan una mayor varianza. Teniendo en cuenta los
resultados en la literatura sobre la ine�ciencia del estimador QML en pequeñas mues-
tras, sería también una ventaja si el tamaño de la muestra es grande cuando se utiliza el
método QML. Por otro lado, la aplicación de la estimación MCL es relativamente más
complicada que la estimación QML. Por lo tanto la estimación MCL requiere mucho más
tiempo para converger. Cuando el numero de series es elevado, las derivadas necesarias
para la estimación de MCL son más difíciles de obtener y si uno desea utilizar derivadas
numéricas, en este caso calcular las derivadas para vectores de estado de gran dimensión
podría resultar muy lento e inestable numéricamente. A la esitmaicón QML no le afecta
en tan gran medida el elevado número de series o el elevado tamaño de la muestra. Por
lo tanto basándonos en nuestra experiencia, sugerimos utilizar el método MCL en la esti-
12
mación de modelos MSV para varias series, como por ejemplo para modelar los retornos
de los índices bursátiles internacionales, y el método de QML podrían ser utilizados para
la estimación de un número mediano a elevado de series, por ejemplo, para los acciones
que cotizan en un mercado de valores.
Capítulo III: Do Correlated Markets Have More VolatilitySpillovers?
En este capítulo se propone un nuevo modelo GARCH para explicar las interacciones
entre las volatilidades de las series de retorno. La motivación proviene de la literatura
de teoría de redes: si una persona tiene el hábito de fumar, es probable que los amigos
cercanos de esta persona también fuma. Además, si los amigos cercanos de una persona
son fumadores, entonces es probable que esa persona sea un fumador también. De mismo
modo, cuando un activo tiene una alta volatilidad, es probable que otro activo que está
altamente correlaciónado con ese activo importe parte de esta volatilidad. Llamamos
a este nuevo modelo NETWORK (NET) GARCH. En este capítulo se explica cómo
estimar este modelo a través del método de máxima verosimilitud.
En la literatura, modelos como BEKK-GARCH de�nidos en Engle y Kroner (1995)
y ECCC-GARCH en Jeantheau (1998) se han usado frecuentemente para estimar las
transmisiones de volatilidad. El modelo BEKK-GARCH tiene la virtud de ser muy
general, ya que permite las transmisiones entre las volatilidades y las covolatilidades. Sin
embargo, como se explica en Bauwens (2006), los parámetros del modelo BEKK-GARCH
son difíciles de interpretar. El modelo ECCC-GARCH de Jeantheau (1998) tiene una
estructura más sencilla, ya que utiliza la descomposición de la matriz de covarianza a
las varianzas condicionales y la matriz de correlación. En ambos modelos el número
de parámetros para estimar aumenta rápidamente con el número de series. Por tanto,
cuando se tienen en cuenta altos números de series la estimación de estos modelos resulta
muy difícil. El número de parámetros del modelo NET-GARCH incrementa linealmente
con el número de series y por lo tanto con elevado números de series este modelo es todavía
estimable. Dado que el modelo NET-GARCH es similar al modelo ECCC-GARCH,
las restricciones de positividad, estacionaridad e identi�cación se pueden derivar de las
condiciones en Jeantheau (1998). Tomamos la ecuación de la volatilidad del modelo
ECCC-GARCH y suponemos las correlaciones dinámicas a través del modelo cDCC-
GARCH, y nos referimos a este modelo con el nombre EcDCC-GARCH. Finalmente,
comparamos el rendimiento del modelo NET-GARCH con el de los modelos EcDCC-
GARCH y cDCC-GARCH.
En este trabajo comprobamos el rendimiento y la validez empírica del modelo NET-
GARCH. Para ello utilizamos los retornos de las acciones que cotizan en el índice FTSE-
13
100 para el período 28/06/2006 - 24/01/2012. También incluimos el índice FTSE-100
en estos datos.
En primer lugar, para demostrar que los parámetros, las volatilidades y las correla-
ciones generadas por el modelo NET-GARCH se estiman correctamente por el método de
máxima verosimilitud, llevamos a cabo experimentos de Monte Carlo, donde generamos
y estimamos los datos por este modelo. Los valores de los parámetros para generar los
datos obtenemos mediante el ajuste del modelo NET-GARCH usando las tres primeras
series y también tres vectores trivariados de series seleccionados al azar de los datos.
En estos cuatro experimentos realizados, los parámetros de volatilidad del modelo NET-
GARCH se estiman con pequeños valores de la raíz del error cuadratico medio. Las
volatilidades y correlaciones se estiman también con bastante precisión.
En la siguiente sección, comparamos el rendimiento de los modelos EcDCC, NET
y cDCC-GARCH en la estimación de Valor en Riesgo (VaR). Después de ordenar los
datos obtenidos a partir de FTSE-100 por orden alfabético, ajustamos estos tres mod-
elos a las primeras dos, tres, cuatro etc. series y calculamos la estimación del VaR de
la cartera de mínima varianza. Como nota Engle y Sheppard (2001), con la cartera
de mínima varianza, la incorrecta especi�cación del modelo tiene un mayor impacto en
las estimaciones. Por otra parte, seguimos Engle y Manganelli (2000) y Engle y Shep-
pard (2001) para probar los rendimientos de los modelos en la estimación de los VARs.
Obtenemos que las estimaciones de VaR del modelo NET-GARCH son muy similares
a las de los modelos EcDCC y cDCC-GARCH. De acuerdo con los resultados de las
pruebas, a pesar de que el modelo NET-GARCH funciona peor que EcDCC-GARCH
en la estimación del VaR, supera al modelo cDCC-GARCH. Dado que en el caso del
modelo EcDCC-GARC el número de parámetros aumentan rápidamente con el número
de series, el model NET-GARCH parece ser una alternativa razonable, ya que captura
las transmisiones de volatilidad con menor numero de parámetros.
Una cuestión que surge es cómo se comporta el modelo NET-GARCH en el caso de
una incorrecta especi�cación del modelo. Para responder a esta pregunta generamos
los datos con el modelo BEKK-GARCH y estimamos con los modelos EcDCC, NET y
cDCC-GARCH. Consideramos un número de series de tres, ocho y trece. Los valores de
los parámetros se han tomado de la estimación de las primeras tres, ocho y trece series
con el modelo BEKK-GARCH. Despues nos �jamos en las estimaciones obtenidas de la
volatilidad de carteras de minima varianza (MVP) y de igual ponderación (EWP) por
los tres modelos. En todos los casos considerados, EcDCC-GARCH supera a los otros
dos modelos en la estimación de las volatilidades. Nuestros resultados sugieren que con
el EWP, el modelo NET-GARCH se comporta mejor que cDCC-GARCH y es similar a
EcDCC-GARCH en la estimación de la volatilidad, mientras que se comporta peron con
14
el MVP. Repetimos el experimento de incorrecta especi�cación mediante la generación
de los datos por el modelo EcDCC-GARCH y la estimación por los modelos EcDCC,
NET y cDCC GARCH. En este caso, para el EWP obtenemos de resultado que NET-
GARCH en realidad funciona aproximadamente como el modelo EcDCC-GARCH, con
lo que no hay errores de especi�cación, mientras que para el MVP tenemos resultados
similares como en el experimento con BEKK-GARCH. Llegamos a la conclusión en esta
sección que, cuando se considera EWP, el modelo NET-GARCH se aproxima al modelo
EcDCC-GARCH y en cambio con el MVP su rendimiento se desvía de el de EcDCC-
GARCH porque el modelo NET-GARCH podría estar sobreestimando la volatilidad de
la serie con una varianza baja.
Teniendo en cuenta que los modelos BEKK-GARCH y EcDCC-GARCH tiene muchos
parámetros, el ajuste de estos modelos en una base de datos de series temporales con
elevado numero de series resulta muy difícil. El modelo NET-GARCH demuestra ser muy
útil en este caso, ya que es capaz de capturar los efectos de transmisiones de volatilidad
con un número relativamente menor de parámetros, lo que implica que, con elevado
número de series este modelo puede ser estimado. Además, como nuestros resultados
sugieren, el rendimiento de este modelo se aproxima al del modelo EcDCC-GARCH. Bajo
la luz de las conclusiones de este capítulo, podemos concluir que el modelo NET-GARCH
es una alternativa razonable al modelo EcDCC-GARCH para capturar los efectos de
transmisiones de volatilidad. A pesar de que con un pequeño número de series, los
modelos BEKK o EcDCC-GARCH son preferidas, ya que ofrecen una estructura de
varianza más ricas, con un elevado número de series, la estimación de estos modelos
podría llegar a ser difícil, cuando sea factible. El modelo NET-GARCH demuestra ser
útil en este caso, ya que requiere la estimación de relativamente un menor número de
parámetros.
En resumen, en esta tesis hemos analizado los rendimientos en pequeñas muestras de
diferentes estimadores de los modelos de volatilidad explicados en este trabajo. Hemos
comparado en el primer capítulo el rendimiento de los estimadores de los parámetros en
una etapa y en múltiples etapas con errores Gaussianas o de Student-t por varios mod-
elos de correlación condicional GARCH y en el segundo capítulo, hemos comparado los
rendimientos de los estimadores cuasi-máxima verosimilitud y verosimilitud de Monte
Carlo en las estimaciones de varios modelos de volatilidad estocástica multivariante.
Además, en esta tesis proponemos modelos multivariantes de volatilitidad para capturar
algunos de los hechos establecidos de los datos de series temporales. En el segundo capí-
tulo, proponemos dos modelos multivariantes de volatilidad estocástica, uno siendo más
general, para capturar el efecto apalancamiento y explicamos cómo estimar estos mode-
los con el método de probabilidad de cuasi-máximo verosimilitud y proporcionamos las
15
transformaciones necesarias para estimar estos modelos con el método de verosimilitud
de Monte Carlo . En el tercer capítulo, proponemos el modelo Network GARCH que
captura los efectos de transmisiones de volatilidad con relativamente menor número de
parámetros en comparación con los modelos BEKK y EcDCC-GARCH.
Los resultados de esta tesis ayudan a comprender las circunstancias en las que utilizar
un método de estimación o otro sirve mejor para el propósito de ajustar un modelo a
los datos. Por otro lado, los modelos propuestos en esta tesis demuestran ser útiles para
captar las regularidades observadas en los datos de series de tiempo.
16
Introduction
Many �nancial time series, such as asset returns or exchange rates, exhibit common
regularities like time varying volatilities or comovements. These similarities are known
in the literature as stylized facts. A desirable time series model is the one that explains
one or more of these regularities. Some of these stylized facts are the following:
� Volatility clustering
It is observed that periods of high (low) volatility are followed by periods of high
(low) volatility. This kind of persistent behavior suggests that there might be an
autoregressive structure governing the dynamics of the volatilities. The ARCH
models introduced by Engle (1982) and SV models introduced by Taylor (1986)
and their extensions referred to throughout this thesis are developed to mimic this
volatility clustering.
� Thick tails
It is documented in Mandelbrot (1963) and Fama (1963, 1965) among others that
asset returns tend to present a heavy tailed or leptokurtic distribution. To match
this stylized fact di¤erent distributions are used in the literature such as Student-
t distribution (see for example Fiorentini et al. 2003, Sandmann and Koopman
(1998)). The �rst chapter of this thesis works with Student-t distribution as well.
� Leverage e¤ects
Leverage e¤ect refers to the negative correlation between the returns and volatili-
ties: i.e. a negative return is expected to increase volatility more than a positive
return. The intuition is that a decrease in the stock prices implies higher leverage
of the �rms, which increases the risks and uncertainty, hence causes high volatility.
Examples can be found in Nelson (1991) or Jungbacker and Koopman (2005). In
the second chapter of this thesis, two multivariate models are proposed to capture
leverage e¤ects.
� Comovements
The volatility spillovers and correlations between returns have been increasingly
of interest in the literature. Among many others Jeantheau (1998), Longin and
Solnik (1995), Bae and Karolyi (1994) are examples analyzing theoretically or
empirically the volatility spillovers. Bollerslev (1990), Engle (2002), Tse (2000),
Pelletier (2006) are some of the papers studying the correlations between returns
of stock markets. The �rst chapter of this thesis refers to the model of Jeantheau
17
(1998) and the third chapter proposes a model to capture volatility spillovers. In
all chapters of this thesis constant and/or time varying correlation models are
considered.
To explain the time varying volatilities, Engle (1982) and Bollerslev (1996) pro-
posed generalized autoregressive conditional heteroskedasticity (GARCH) models. In
GARCH set up, the volatilities follow a deterministic equation of the squared previous
day returns and volatilities. Therefore the dynamics of the volatilities in this model
is observation driven. Later, the GARCH models have been extended to multivariate
settings (MGARCH) to capture the volatility spillovers and correlations between series,
see for example, Bauwens et al. (2006) and Silvennoinen and Teräsvirta (2009) for a
survey. Among others, for example EGARCH proposed by Nelson (1991) is developed
to explain the leverage e¤ects, ECCC-GARCH of Jeantheau (1998) is developed for
capturing the volatility spillovers, Bollerslev (1990) proposes the constant conditional
correlation GARCH (CCC-GARCH) model capturing the correlation between two series
and Engle (2002) proposes the DCC-GARCH model to allow these correlations between
returns to vary over time.
Alternatively, the stochastic volatility (SV) literature started by Taylor (1986, 1994)
and Hull andWhite (1987) suggests to model the time varying volatility as an unobserved
component and lets its logarithm follow an autoregressive process. In this set up, the
volatilities are parameter driven. Starting with the constant correlations multivariate
stochastic volatility (CC-MSV) of Harvey et al. (1994), the multivariate extensions of
the SV method have been developed. Among others, Asai and McAleer (2006) proposes
the MSV with leverage model to explain the leverage e¤ects in a time series data, Harvey
et al. (1994) proposes the CC-MSV model to capture the correlation between the returns
of a time series data while Jungbacker and Koopman (2006) proposes the time varying
correlations MSV model to allow these correlations between returns to change over time.
The SV approach is attractive in the sense that it is closer to the models used in the
�nancial theory to describe the behavior of prices; see Shephard and Andersen (2008).
Moreover it has been shown that the SV models describe the behavior of volatilities more
accurately compared to GARCH models; see Danielsson (1994), Kim et al. (1998) and
Carnero et al. (2004). Although statistically more attractive than the GARCH models,
SV models have the disadvantage in terms of estimation because the exact likelihood
functions of these models are di¢ cult to evaluate.
In this thesis, we have worked with both multivariate GARCH and SV models. The
objective of this thesis is to study the properties of several methods that estimate volatil-
ity models and to propose new models to explain one or more of the stylized facts men-
tioned above. In particular, the focus of the three chapters have been on the multivariate
18
modelling and estimation of the time varying volatility. In the �rst chapter we analyze
the small sample performance of the multiple steps estimators in the case of Gaussian
and Student-t errors. For this reason we performed simulation experiments where di¤er-
ent parameter values and settings are considered. Moreover we checked the robustness of
our results with respect to the distribution and model assumptions. Finally we consider
the case where the true distribution is skewed normal or Student-t but the distribution
assumed in the estimation is symmetric. In the second chapter we compare the per-
formances of two method of estimation, namely the quasi-maximum likelihood (QML)
of Harvey et al. (1994) and Monte Carlo likelihood (MCL) of Jungbacker and Koop-
man (2006), of multivariate SV (MSV) models. We also propose two MSV models with
leverage, one being more general in the sense that it allows for the correlation between
the volatility shocks of series i with the return shocks of series j. We analyze the per-
formances of the QML and MCL methods in estimating the constant correlations (CC)
MSV of Harvey et al. (1994), time varying correlations (TVC) MSV of Jungbacker and
Koopman (2006) and the two MSV models with leverage that we propose. In the third
chapter we propose a new multivariate GARCH model, which we refer to as the Network
(NET) GARCH model, which can capture the volatility spillovers with relatively small
number of parameters and we compare its performance with other existing multivari-
ate GARCH models that allow for volatility spillovers. We check if the NET-GARCH
model�s parameters can be estimated well by the maximum likelihood method explained
in the paper. Later we look at the performance of this model and the other models in
estimating the Value at Risk (VaR) in the data. Finally we check how the NET-GARCH
model performs when the model is misspeci�ed. Even though this thesis consists of three
chapters, all throughout the thesis, the focus is on the multivariate modelling and es-
timation of volatility models. In the �rst and second chapters the interest is on the
estimation methods of di¤erent multivariate volatility models while in the second and
third chapters new models are proposed.
In all the chapters of the thesis, the plan of study consisted of explaining theoretical
and econometric backgrounds of the models and estimation methods and performing sim-
ulation experiments to analyze the small sample performance of the models or methods
of question. The results of the simulation experiments are reported in tables via means,
standard errors and root mean squared errors. Moreover, we include some �gures to
plotting the kernel density estimates for the di¤erences between the estimated and the
true values of the volatilities and correlations. Finally in each chapter, an empirical
estimation is included for illustration purposes using the data from stock markets. All
the programming and implementation of the models and estimation methods are done
in MATLAB.
19
Next, a brief summary of the three chapters is presented:
Chapter I: Estimating VAR-MGARCH Models in MultipleSteps
In this chapter we analyze the small sample performance of the multiple steps es-
timation of vector autoregressive multivariate GARCH models (VAR-MGARCH). For
this purpose, we consider �ve conditional correlation GARCH models given by di¤erent
speci�cations of the correlation dynamics: the constant conditional correlation (CCC)
GARCH of Bollerslev (1990), dynamic conditional correlation (DCC) GARCH of Engle
(2002), consistent DCC (cDCC) GARCH of Aielli (2008), extended conditional corre-
lation (ECCC) GARCH of Jeantheau (1998) and regime-switching dynamic correlation
(RSDC) GARCH of Pelletier (2006). For the estimation of DCC and cDCC-GARCH
models we use the correlation targeting approach (see Engle 2009) which reduces the
number of parameters to estimate by replacing the long run covariance matrix by the
sample covariance matrix. Aielli (2008) shows that estimation of DCC-GARCH models
with correlation targeting results in inconsistent estimators and he suggests a corrected
version of DCC-GARCH model we refer to as cDCC-GARCH model. On the other
hand, it is not uncommon to encounter dynamics in the conditional mean which can be
modelled with a VARMA model (see for example Veiga and McAleer 2008a, 2008b). For
simplicity we assume a VAR(1) speci�cation for the conditional means.
One way to estimate the parameters of all these VAR-MGARCH models is to max-
imize the full log-likelihood. Assuming that there is no misspeci�cation of the error
distribution, the resulting estimators are maximum likelihood (ML) estimators and are
consistent and asymptotically normal. If the true distribution is not Gaussian but the
estimation is based on Gaussian errors, then the estimators are quasi-maximum likeli-
hood estimators (QML) and they are also consistent and asymptotically normal. See
for example Bollerslev and Wooldridge (1992). If the estimation is based on Student-t
errors, then as long as the true distribution is symmetric, the QML estimators will be
consistent and asymptotically normal. See Newey and Steigerwald (1997).
An alternative to the full maximization of the likelihood is using multiple steps es-
timators. Under the Gaussianity assumption, we can estimate the parameters in two
steps; see for example Engle (2002). In a �rst step we can estimate the mean and vari-
ance parameters and taking the estimates of these parameters as given in a second step
we estimate the correlation parameters. As Engle and Sheppard (2001) suggests for
the DCC-GARCH model, the two-step estimators will be consistent and asymptotically
normal but ine¢ cient.
20
Finally, we could estimate the parameters of these models in three steps: �rst we
estimate the mean parameters, then taken mean parameter estimates as given, we can
estimate the variance parameters and then taking the mean and variance parameter
estimates as given we can estimate the correlation parameters. This three-step estimation
method is mentioned in Bauwens et al. (2006). This method could be seen as a two step
estimation for a zero mean series. Therefore three-step estimators are also consistent
and asymptotically normal but ine¢ cient.
Although not supported by theory, we also consider multiple steps estimation based
on Student-t distribution, where in all the steps the estimation is based on this distri-
bution. Bauwens and Laurent (2005) and Jondeau and Rockinger (2005) also analyze
multiple steps estimators based on Student-t distribution; however they estimate the
variance parameters assuming Gaussian errors while estimating the correlation parame-
ters assuming Student-t errors.
In this chapter we perform several Monte Carlo experiments to analyze and compare
the small sample performance of one-step, two-steps and three-steps estimators of the
�ve models mentioned above assuming Gaussian or Student-t errors.
We �nd out that when the estimation is based on normal distribution, the perfor-
mance of one-step and multiple steps estimators are very similar. However, when the
estimation is based on Student-t distribution, for some models the di¤erences between
the estimators could be relevant. Therefore multiple steps estimation based on Student-t
distribution may not be a good idea.
We also checked the robustness of our results with respect to misspeci�cation of the
error distribution or the model. Our results show that if the true error distribution is
Student-t but estimation is based on the Gaussian distribution, kernel density estimates
of the estimates of volatility and correlation obtained from one-step and multiple steps
estimators are quite similar. Analogously, if the true error distribution is Gaussian but
estimation is based on the Student-t distribution, we obtain the same results as when
the true and assumed distribution is a Student-t. When the model is misspeci�ed and
the estimation is based on Gaussian errors, we �nd that, on average, volatilities and
correlations are relatively well estimated even when using a misspeci�ed model. The
multiple-steps estimates of volatilities (correlations) deviate from the true values at most
by 2 % more than what one-step estimates of the correctly speci�ed model do.
When errors are distributed as a skewed Student-t but the estimation is performed
assuming non-skewed Gaussian or Student-t errors, we �nd that kernel density estimates
of the di¤erence between one-steps and multiple steps estimates of volatilities and corre-
lations from their true values are very similar when the estimation is based on a Gaussian
distribution. However, this is not true when the estimation is based on Student-t errors.
21
In any case, when the true distribution is skewed, one should be cautious in using one-
step or multiple-steps estimators based on Student-t errors since both are inconsistent
estimators.
Chapter II: Estimation of Multivariate Stochastic VolatilityModels: A Comparative Monte Carlo Study
In stochastic volatility models the time varying volatilities are considered as an unob-
served component such that the log-volatilities are let to follow an autoregressive process.
As we mentioned above, although stochastic volatility models are statistically more at-
tractive, they are more costly in terms of implementation because their exact likelihood
functions are di¢ cult to evaluate.
Harvey et al. (1994) proposed the quasi-maximum likelihood (QML) method to esti-
mate the multivariate SV models. Their method takes the log-squared transformations of
the observations. The distribution of the log-squared errors in the observation equation
is approximated by a normal distribution, therefore this method is based on approxi-
mations. If we take the log-volatilities as a state equation, then we obtain a Gaussian
state space form equation. After this, a Kalman �lter is used to construct the prediction
error decomposition of the likelihood which is maximized to obtain the estimators. Ruiz
(1994) shows that this QML method is consistent and asymptotically normal. QML
estimation is very easy to implement, however it is ine¢ cient given that it is based on
approximations. See Jacquier et al. (1994), Breidt and Carriquiry (1996) and Sandmann
and Koopman (2005) as examples.
The evaluation of the exact likelihood function requires high dimensional integration
which could be obtained by simulation methods. One method that achieves this is the
Monte Carlo likelihood (MCL) estimation method of Jungbacker and Koopman (2006).
This method does not use a log-squared transformation of the observations. As Durbin
and Koopman (1997) showed, the log-likelihood of the state space models with non-
Gaussian errors can be written as a sum of the log-likelihood of the approximating
Gaussian model and a correction for the departures from the Gaussian assumption with
respect to the true model. What makes this simulated maximum likelihood method more
attractive than others is that the simulations are only needed to evaluate the correction
part in the log-likelihood. Therefore it requires less number of simulations to evaluate
the log-likelihood and takes much less time to do so. MCL method of Jungbacker and
Koopman (2006) is consistent, asymptotically normal and e¢ cient. It should be noted
that MCL method might require speci�c transformations to estimate some MSV models,
but as long as these transformations can be done MCL method proves to be very useful
in obtaining asymptotically e¢ cient estimators. Also MCL method entails calculation
22
of derivatives with respect to the state variables, which could be obtained analytically
or numerically.
In this chapter we compare the small sample performance of QML method of Harvey
et al. (1994) and MCL method of Jungbacker and Koopman (2006) in estimating the
Constant Correlation (CC) MSV of Harvey et al. (1994), Time Varying Correlation
(TVC) MSV of Jungbacker and Koopman (2006) and for two MSV models with leverage
we propose. Asai and McAleer (2006) also proposed an MSV model with leverage, where
the leverage matrix is diagonal. This implies that the future volatility shocks of series
i is only correlated with the return shocks of series i. Moreover, these correlations are
proportional to the variances of the volatility shocks, which is a restriction imposed on
the model. In this chapter, we propose an MSV model with diagonal leverage where
there is no such restriction on the correlation. Also, to allow for the possible correlation
between the volatility shocks of series i and return shocks of series j, we propose the MSV
model with non-diagonal leverage. For the estimation of these two models with QML
method, we adapt the transformations given in Asai and McAleer (2006). To estimate
them with MCL method, we provide the necessary transformations.
Based on our results, we conclude that even though in the case of TVCMSV, QML
estimator performs close to MCL estimator, the latter is always preferred. We do not
recommend using QML estimators for the models with leverage. Although QML method
can be implemented much easier than MCL and the estimation time is much less in QML
estimation; we suggest its use if it is expected that the series have high and/or time
varying correlation and the SV processes have higher variance. Given the results in the
literature on the ine¢ ciency of QML estimator in small samples, it would be also a plus if
the sample size is large, when using QMLmethod. On the other hand the implementation
of MCL estimation is relatively more complicated than the QML estimation. Therefore
MCL estimation requires much more time to converge. When the cross-section size is
large, the analytical derivatives for the MCL estimation are harder to obtain and if one
would like to use numerical derivatives in this case, then the derivatives calculated for
large state vectors could be very time consuming and numerically unstable. QML is not
as much a¤ected by the large cross-sections or large sample sizes. Therefore based on
our experience, we would suggest using MCL method in the estimation of MSV models
for several series, as for modelling the returns of international stock market indices, and
QML method could be used for the estimation with medium-to-large number of series
from a stock market.
Chapter III: Do Correlated Markets Have More VolatilitySpillovers?
23
In this chapter we propose a new GARCH model to explain the volatility interaction
between return series. The motivation comes from Network Theory literature: if a person
has smoking habits, it is likely that the close friends of this person also smoke. Also,
if close friends of a person are smokers, then that person is likely to be a smoker as
well. Similarly, when an asset i is experiencing high volatility, an asset j that is highly
correlated with (closely related to) asset i is likely to import some of this volatility. We
name this new model Network (NET) GARCH. We discuss how to estimate this model
via maximum likelihood method.
In the literature, BEKK-GARCH as de�ned in Engle and Kroner (1995) and ECCC-
GARCH model of Jeantheau (1998) have been commonly used to estimate volatility
spillovers. BEKK-GARCH model has the virtue of being very general because it allows
for spillovers between volatilities and covolatilities. However as noted as well noted in
Bauwens (2006), the parameters of BEKK-GARCH model are hard to interpret. ECCC-
GARCH model of Jeantheau (1998) has a simpler structure because it uses the decom-
position of the covariance matrix to conditional variances and the correlation matrix. In
both of these models the number of parameters to estimate increases rapidly with the
number of series. Therefore when high cross-sections are considered, the estimation of
these models become practically very di¢ cult, if feasible. The number of parameters
of NET-GARCH model increases linearly with the number of series therefore with high
cross-sections this model is still estimable. Given that NET-GARCH model is similar to
(but not nested by) the ECCC-GARCH model, the positivity, stationarity and identi�-
cation restrictions can be derived from the conditions in Jeantheau (1998). We take the
volatility equation of ECCC-GARCHmodel and assume dynamic correlations via cDCC-
GARCH model, and we refer to this model as EcDCC-GARCH model. We compare the
performance of NET-GARCH model with that of EcDCC-GARCH and cDCC-GARCH
models.
In this chapter we check the performance and empirical validity of the NET-GARCH
model. For this purpose we use the returns of the stocks listed in FTSE-100 for the
period 28/06/2006 - 24/01/2012. We also include the FTSE-100 index to this data.
First, to show that parameters, volatilities and correlations generated by the NET-
GARCH model is estimated well by the maximum likelihood method, we perform Monte
Carlo experiments where the data is generated and estimated by this model. The true
parameter values are obtained by �tting a NET-GARCH model to the �rst three series
and also to the estimation of three trivariate series randomly selected from the data. In
all four experiments, the volatility parameters of the NET-GARCH model are estimated
with small root mean squared errors. The volatilities and correlations were also estimated
quite accurately.
24
In the next section, we compare the performance of EcDCC, NET and cDCC-GARCH
models in estimating Value at Risk (VaR). After ordering the data obtained from FTSE-
100 alphabetically, we �t these three models to the �rst two, three, four etc. series
and calculate the VaR estimate for the minimum variance portfolio. As Engle and
Sheppard (2001) note, with minimum variance portfolio the model misspeci�cation has
higher impact on the estimates. Furthermore, we follow Engle and Manganelli (2000)
and Engle and Sheppard (2001) to test the model�s performance in estimating the VaRs.
We �nd out that the NET-GARCH model�s VaR estimates are very similar to those
of the EcDCC and cDCC-GARCH models. According to the test results, although
NET-GARCH model performs worse than EcDCC-GARCH in estimating the VaRs, it is
outperforming the cDCC-GARCH model. Given that the number of parameters increase
rapidly with the number of series in case of EcDCC-GARCHmodel, NET-GARCHmodel
seems to be a reasonable alternative as it captures the volatility spillovers with much
less number of parameters.
One question that could arise is how the NET-GARCH model behaves in case of a
model misspeci�cation. To answer this question we generate data with BEKK-GARCH
model and estimate with EcDCC, NET and cDCC-GARCH models. We consider the
cross-section sizes three, eight and thirteen. The values of the parameters are taken from
estimating the �rst three, eight and thirteen series with BEKK-GARCHmodel. Later we
look at the volatility estimates obtained by the three models for the minimum variance
(MVP) and equally weighted (EWP) portfolios. In all the cases considered, EcDCC-
GARCH outperforms the other two models in estimating the volatilities. Our results
suggest that with EWP, the NET-GARCH model is performing better than cDCC-
GARCH and close to EcDCC-GARCH models in estimating the volatilities while with
MVP it is not doing very well. We repeat the misspeci�cation experiment by generating
the data by EcDCC-GARCH model and estimating with EcDCC, NET and cDCC-
GARCH models. In this case, for the EWP we �nd out that NET-GARCH is actually
performing very closely to the EcDCC-GARCH model, for which there is no misspeci�-
cation, while we have similar results as in the experiment with BEKK-GARCH for the
MVP. We conclude in this section that when EWP is considered NET-GARCH model
is actually performing very closely to the EcDCC-GARCH model even though it could
be overestimating the volatilities of the series with low variance and therefore deviating
from the EcDCC-GARCH estimates with MVP.
Given that BEKK-GARCH and EcDCC-GARCH models have many parameters, �t-
ting these models to a time series data with high cross-section size becomes very di¢ cult.
NET-GARCH model proves to be very useful in this case because it is able to capture
the volatility spillovers with relatively less number of parameters, which imply that with
25
large cross-sections as well this model can be estimated. Moreover as our results sug-
gest, the performance of this model approaches to the performance of EcDCC-GARCH
model. Under the light of the �ndings in this chapter, we can conclude that NET-
GARCH model is a reasonable alternative to the EcDCC-GARCH model to capture the
volatility spillovers. Even though with small number of series BEKK or EcDCC-GARCH
models should still be preferred because they o¤er a richer variance structure, with high
number of series the estimation of these models could become di¢ cult, when feasible.
NET-GARCH model proves to be useful in this case because it requires estimation of
much less parameters.
To sum up, in this thesis we have analyzed the small sample performance of di¤erent
estimators of the volatility models considered. We compared the small sample perfor-
mance of one-step and multiple steps estimators with Gaussian and Student-t errors in
the �rst chapter for several conditional correlation GARCH models and in the second
chapter we compared the performance of quasi-maximum likelihood and Monte Carlo
likelihood estimators. Moreover we propose multivariate volatility models to capture
some of the stylized facts in the data. In the second chapter, we propose two multi-
variate stochastic volatility models to capture the leverage e¤ects and we discuss how to
estimate these models with quasi-maximum likelihood method and provide the necessary
transformations to estimate these models with the Monte Carlo likelihood method. In
the third chapter, we propose the Network GARCH model which captures the volatility
spillovers with less number of parameters compared to that of the BEKK and ECCC-
GARCH models. The �ndings of this thesis help understand under which circumstances
using one or the other estimation method serves more to the purpose of �tting a model
to the data. On the other hand, the models proposed in this thesis prove to be useful in
capturing the regularities observed in time series data.
26
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30
Chapter 1
Estimating VAR-MGARCH Modelsin Multiple Steps
1.1 Introduction
Understanding how stock market returns and volatilities move over time has been of
interest to researchers into the time series literature. In addition, as the �nancial crisis
has shown, it is also very important to realize that stock markets move together. Evi-
dence of these comovements can be found, for example, in the fall of several international
stock market indices after a very big investment bank in US, Lehman Brothers, declared
bankruptcy in September 2008. Therefore, trying to model stock markets in a univariate
way ignoring their interactions would be insu¢ cient. In this sense, Multivariate General-
ized Autoregressive Conditional Heteroskedasticity (MGARCH) models have been very
popular to capture the volatility and covolatility of assets and markets; see, for example,
Bauwens et al. (2006) and Silvennoinen and Teräsvirta (2009) for a survey.
One of the problems with many MGARCH models is the di¢ culty to verify that the
conditional variance-covariance matrix is positive de�nite. Engle et al. (1984) provide
necessary conditions for the positive de�niteness of the variance-covariance matrix in a
bivariate ARCH setting. However, extensions of these results to more general models
are very complicated. Moreover, imposing restrictions on the log-likelihood function, in
order to have the necessary conditions satis�ed, is often di¢ cult.
A model that could avoid these problems is the Constant Conditional Correlation
GARCH (CCC-GARCH) model proposed by Bollerslev (1990). In this model, the
Gaussian maximum likelihood (ML) estimator of the correlation matrix is the sam-
ple correlation matrix which is always positive de�nite. Therefore, the only restrictions
needed are the ones for the conditional variances to be positive. On top of that, since
31
the correlation matrix can be concentrated out of the log-likelihood function, the opti-
mization problem becomes simpler. Consequently, the CCC-GARCH model has become
very popular in the literature regardless of some limitations such as the constant corre-
lation assumption and the incapability to explain possible volatility interactions. The
extension proposed by Jeantheau (1998), the ECCC-GARCH model, addresses the last
issue by allowing for volatility spillovers. Relaxing the constant correlation assumption
is done by Engle (2002) and Tse and Tsui (2002) who propose the Dynamic Condi-
tional Correlation GARCH (DCC-GARCH) model in which the correlation changes over
time. However, since the correlation dynamics require more parameters, the estimation
of the DCC-GARCH model can be computationally very heavy. One possible solution
is to use the correlation targeting approach, see Engle (2009), in which the intercept in
the correlation equation is replaced by its sample counterpart. This solution is ques-
tioned by Aielli (2008) who suggests a correction to the DCC-GARCH model, denoted
by Consistent DCC-GARCH (cDCC-GARCH) model.
Alternatively, Pelletier (2006) introduces the Regime Switching Dynamic Correlation
GARCH (RSDC-GARCH) model in which the correlation is constant over time but
changing between di¤erent regimes and driven by an unobserved Markov switching chain.
This model can be thought as in between the CCC-GARCHmodel and the DCC-GARCH
model, with the problem that the number of correlation parameters to be estimated
increases rapidly with the number of series considered.
When dealing with stock market returns, it is not unusual to �nd some dynamics in
the conditional mean, that could be well approximated by a Vector Autoregressive Mov-
ing Average (VARMA) model; see, for example, da Veiga and McAleer (2008a, 2008b).
One way to estimate the parameters of the VARMA-MGARCH conditional correlation
model would be solving the optimization problem of the full log-likelihood function and
therefore obtaining the estimates for all the parameters in one step. If a Gaussian
log-likelihood function is speci�ed and the true data generating process (DGP) is also
Gaussian, then it is known that ML estimators are consistent and asymptotically normal.
In the case that the true DGP is not Gaussian, then we would be using quasi-maximum
likelihood (QML) estimators. Bollerslev and Wooldridge (1992) show that, under quite
general conditions, QML estimators are consistent and asymptotically normal. Estimat-
ing all parameters in one step would be the best we could achieve, however when there
are many parameters involved, it is very heavy computationally, when feasible. Boller-
slev (1990), Longin and Solnik (1995) and Nakatani and Teräsvirta (2008) are few of the
papers using one-step estimation.
Under the normality assumption, the parameters could also be estimated in two
steps. First, the mean and variance parameters are estimated assuming no correlation
32
and then, in a second step, the correlation parameters are estimated given the estimates
from the �rst step; see, for example, Engle (2002). However, as Engle and Sheppard
(2001) suggest for the DCC-GARCH model, these two-step estimators will be consistent
and asymptotically normal but not e¢ cient.
The three-steps estimation method is mentioned in Bauwens et al. (2006). It consists
of estimating the mean parameters in a �rst step, the variance parameters in a second
step, given the �rst step estimates, and �nally, given all other parameter estimates, the
correlation parameters in the last step. The second and third steps of the procedure
will be equivalent to the two-steps estimation method for a zero-mean series. Therefore,
under normal errors, the three-steps estimators are also consistent and asymptotically
normal. Engle and Sheppard (2001) implement the three-steps estimation procedure in
the empirical part of their paper.
Evidence gathered over the past decades shows that stock market returns are often far
from having a normal distribution. Consequently, we also consider estimating the models
assuming a Student-t distribution. The one-step estimator is obtained by maximizing
the log-likelihood function based on the multivariate t-distribution; see, for example,
Harvey et al. (1992) and Fiorentini et al. (2003). Although there is no theoretical
work studying the properties of multiple steps estimation when assuming a Student-t
distribution, we consider two-steps and three-steps estimators. In this line of research,
Bauwens and Laurent (2005) and Jondeau and Rockinger (2005) also analyze two-steps
estimators. However, their approach is di¤erent in the sense that the �rst step of their
estimation is performed assuming Gaussian errors while we maintain the assumption
that the errors are distributed as a Student-t.
In this paper, we present various Monte Carlo experiments to compare the �nite
sample performance of the more e¢ cient one-step estimator with the two-steps and
three-steps estimators for di¤erent Vector Autoregressive Multivariate Conditional Cor-
relation GARCH models. In particular we consider VAR(1) - CCC, ECCC, DCC, cDCC
and RSDC - GARCH(1,1) models. When the data is normally distributed, we �nd that,
for the models considered and for the sample sizes usually encountered in �nancial econo-
metrics, di¤erences between the one-step and multiple steps estimators are negligible.
When we change the assumption on the distribution to a Student-t, we conclude that,
for some models, the di¤erences between the estimators could be relevant and therefore,
estimating the parameters in multiple steps might not be a good idea.
The comparison between one-step and two-steps estimators helps us to measure the
e¢ ciency loss when estimating the correlation parameters separately from the mean and
variance parameters; see Engle (2002) and Engle and Sheppard (2001). As we will see,
when the errors are assumed to be Gaussian, the small sample behavior of one-step and
33
two-steps estimators is very similar. On the other hand, when the estimation is based
on the Student-t distribution, in some cases two-steps estimators deviate from one-step
estimators.
Comparing two-steps and three-steps estimators helps us to analyze the e¤ects of
separating the estimation of mean and variance parameters; see Bauwens et al. (2006).
Our results show that, when the errors are assumed to be Gaussian or Student-t, the
small sample behavior of two-steps and three-steps estimators is also very similar.
Some robustness checks have been carried out to study how the results change when
the true error distribution is di¤erent from the assumed one. Also, we analyze the
robustness of our �ndings to the model misspeci�cation.
One potential problem of our results is their external validity. For the Monte Carlo
experiments, we considered bivariate models and in some cases three time series. We
assume that what we �nd for two and three time series could be extrapolated for any
number k > 3 of time series.
The rest of the paper is structured as follows. Section 1.2 introduces the econometric
models of interest. One-step and multiple steps estimators for the previous models are
discussed in Section 1.3. Section 1.4 describes the Monte Carlo experiments and presents
a discussion of the results. Finally, Section 1.5 concludes the paper.
1.2 Econometric Models
For simplicity we consider a k-variate Vector Autoregressive (VAR) model of order one
for the mean equation with the following notation:
Yt = �+ �Yt�1 + "t (1.1)
where V ar("t jYt�1; :::Y1) = Ht, Yt is a k � 1 vector of returns, � is a k � 1 vectorof constants, � is a k� k matrix of autoregressive coe¢ cients and "t is a k� 1 vector oferror terms as follows.
Yt =hy1t y2t : : : ykt
i0; � =
h�1 �2 : : : �k
i0
� =
266664�11 �12 : : : �1k�21 �22 : : : �2k...
.... . .
...
�k1 �k2 : : : �kk
377775 ; "t =h"1t "2t : : : "kt
i0
34
The model is stationary if all values of z solving equation (1.2) are outside of the unit
circle.
jIk � �zj = 0 (1.2)
The number of mean parameters in the coe¢ cient matrices � and � is k(k + 1):
However, sometimes � is assumed to be diagonal. In that case, there will be 2k mean
parameters to estimate.
The error term "t can be written as follows
"t =H1=2t �t
where �t is a k � 1 vector with E(�t) = 0 and V ar(�t) = Ik.
H t = DtRtDt (1.3)
where Dt = diag(h1=21t ; h
1=22t ; :::; h
1=2kt ) and Rt is the conditional correlation matrix such
that
H t = diag(h1=21t ; h
1=22t ; :::; h
1=2kt )
2666641 �12t : : : �1kt�12t 1 : : : �2kt...
.... . .
...
�1kt �2kt : : : 1
377775 diag(h1=21t ; h1=22t ; :::; h1=2kt )
=
266664h1t �12t
ph1th2t : : : �1kt
ph1thkt
�12tph1th2t h2t : : : �2kt
ph2thkt
......
. . ....
�1ktph1thkt �2kt
ph2thkt : : : hkt
377775From previous equations, given that the conditional correlation matrix, Rt, is always
positive de�nite, it is clear that as long as conditional variances, hit, are positive for
any i = 1; 2; : : : ; k, the conditional variance-covariance matrix, Ht, will be also positive
inde�nite. The conditional variances hit are assumed to follow a GARCH(1,1) model.
Then,
ht =W +A"(2)t�1 +Ght�1 (1.4)
where ht =hh1t h2t : : : hkt
i0and "(2)t =
h"21t "22t : : : "2kt
i0are k � 1 vectors of
conditional variances and squared errors respectively and W is a k � 1 and A and
G are k � k matrices of coe¢ cients. If A and G are restricted to be diagonal; see,
for example, Bollerslev (1990) and Engle (2002), then volatility spillovers cannot be
35
captured. Alternatively, if A and G are non-diagonal; see, for example, Jeantheau
(1998) and Ling and McAleer (2003), then the model allows for volatility spillovers. In
the former case there will be 3k variance parameters to estimate, while in the latter that
number will be k(2k + 1).
Let us denote by !i = [W]i, �ij = [A]i;j and ij = [G]i;j. The following conditions,
in Jeantheau (1998), are su¢ cient for the variances to be always positive.
!i > 0 �ij > 0 ij > 0 for all i and j:
Nakatani and Teräsvirta (2008) provide necessary and su¢ cient conditions for ht to have
positive elements for all t. They show that o¤-diagonal elements in G could be negative
while Ht is still positive de�nite. This allows for negative volatility spillovers; see also
Conrad and Karanasos (2010). The model is stationary in covariance if the roots of
jIk � (A+G)zj = 0 are outside of the unit circle. In the diagonal case, this conditionis equivalent to
�ii + ii < 1 for all i:
This paper considers �ve conditional correlation GARCH models given by di¤erent
speci�cations of Rt in (1.3). The �rst and simplest one is the CCC-GARCH model
where the correlations are restricted to be constant over time. Bollerslev (1990) shows
that, under this restriction, the Gaussian ML estimator of the correlation matrix, Rt =
R, is equal to the matrix of sample correlations of the standardized residuals, i.e.
[bR]ij = b�ij = Pt b�itb�jtq�P
t b�2it� �Pt b�2jt� (1.5)
where �t = D�1t "t are the standardized errors. Notice that, in this case, the number of
correlation parameters to be estimated is only k(k�1)=2. The ECCC-GARCH model
of Jeantheau (1998) extends the CCC-GARCH model by allowing for volatility spillovers
as A and G in (1.4) are non-diagonal.
The third model we consider is the DCC-GARCH in which Rt = PtQtPt with
Pt = diag(Qt)�1=2 and Qt = (1� �1� �2)Q+ �1�t�1� 0t�1+ �2Qt�1 where Qt denotes the
covariance matrix and Q is the long run covariance (correlation) matrix. The correlation
targeting approach suggests replacing Q with the sample covariance matrix of the stan-
dardized errors �t; see Engle (2009). This procedure makes the estimation easier since it
reduces the number of correlation parameters from k(k � 1)=2 + 2 to only 2: �1 and �2.If both are non-negative scalars satisfying �1 + �2 < 1, then the correlation matrix, Rt;
will be positive de�nite. Hafner and Franses (2009) provide a more general de�nition of
the model where they consider coe¢ cient matrices instead of scalar coe¢ cients allow-
ing for di¤erent dynamics on di¤erent correlations. However, this increases the number
36
of parameters considerably. For simplicity, we will focus on the set up with the scalar
coe¢ cients.
The DCC-GARCH model su¤ers from two problems. First, as Engle and Sheppard
(2001) and later Engle, Shephard and Sheppard (2008) point out, when k is large the
correlation targeting approach used in the DCC-GARCH model causes signi�cant bi-
ases to estimators of the parameters �1 and �2. To �x this problem, Engle, Shephard
and Sheppard (2008) suggest a composite likelihood estimator which is based on the
sum of the likelihoods obtained from smaller number of series and therefore avoid the
trap of high dimensionality. Another solution is proposed by Hafner and Reznikova
(2010), where the authors use shrinkage to target methods to eliminate these biases
asymptotically. The second problem, as Aielli (2008) argues, is that multiple steps es-
timators of DCC-GARCH models with correlation targeting are inconsistent since the
covariance matrix of the standardized residuals is not a consistent estimator of the long
run covariance matrix Q. As Caporin and McAleer (2009) point out as well, Aielli�s
conclusion follows from the fact that the unconditional expectations of Qt could di¤er
from the unconditional expectation of �t�1� 0t�1, the former being a covariance matrix
while the latter is a correlation matrix by construction. Aielli (2008) therefore suggests
a corrected version of the DCC-GARCH model, denoted by cDCC-GARCH, in whichQt = (1 � �1 � �2)Q + �1�
�t�1�
�0t�1 + �2Qt�1 where ��t = diag(Qt)
1=2�t. He argues that
in this model a natural estimator for the long run covariance matrix, Q, would be the
sample covariance matrix of ��t . The number of parameters to be estimated will be then
only 2 as in the DCC-GARCH model of Engle (2002).
The last model we will consider in this paper is the RSDC-GARCH. In this modelthe conditional correlations follow a switching regime driven by an unobserved Markov
chain such that they are �xed in each regime but may change across regimes. For simplic-
ity, we assume a two-states Markov process such thatRt, at any time t, could be equal to
either RL or RH , which stands for low and high state correlation matrices, respectively.
The transition probabilities matrix is given by � = ff�L;L; �H;Lg; f�L;H ; �H;Hgg, where�i;j is the probability of moving from state j to state i. Given that �j;j + �i;j = 1, the
number of correlation parameters is k(k � 1) + 2:In the next section we will discuss how to estimate the parameters of these models.
1.3 Estimation Procedures
Multivariate GARCH models can be estimated using maximum likelihood. However,
how the estimation is implemented in practice is one of the main problems. When the
number of parameters is large, it is common that optimization procedures fail to �nd
37
the maximum of the likelihood function. In this section we will describe alternative
estimation methods which could be used in practice.
Let us start by introducing some notation. Let � = (�0; vec(�)0)0 be the vector
containing all the mean parameters in equation (1.1). The vector containing all the
variance parameters in (1.4) will be denoted by � = (W0; vec(A)0; vec(G)0)0 and will
be the one with all the correlation parameters, that will change according to the model
considered in each case. For example, = vech(R) for a CCC-GARCH model, while
for a cDCC-GARCH model, it will be = (vech(Q)0; �1; �2)0:1
1.3.1 Vector Autoregressive CCC, ECCC, DCC and cDCCGARCHmodels
In this section we analyze three possible procedures to estimate the parameters in equa-
tions (1.1) and (1.3), denoted by � = (�0; �0; 0)0 when Rt in equation (1.3) is speci�ed
by the CCC-GARCH, ECCC-GARCH, DCC-GARCH or the cDCC-GARCH model.
1.3.1.1 One-step Estimation
One possibility is to estimate all parameters of the model, � = (�0; �0; 0)0 simultane-
ously. If data is assumed to be normally distributed, this one-step estimator will be the
maximum likelihood estimator of � and it can be found by maximizing the multivariate
Gaussian log-likelihood function:
L(�) = �Tk2log(2�)� 1
2
TXt=2
(log jHtj+ "0tH�1t "t)
From equation (1.3) we have that
L(�) = �Tk2log(2�)� 1
2
TXt=2
log jDtRtDtj �1
2
TXt=2
"0t(DtRtDt)�1"t =
= �Tk2log(2�)� 1
2
TXt=2
log jRtj �TXt=2
log jDtj �1
2
TXt=2
� 0tR�1t �t (1.6)
If errors are assumed to follow a Student-t distribution, then the function to be
maximized will be the multivariate Student-t log-likelihood as in Fiorentini et al. (2003):
1Notice that the vec operator stacks the colums of a matrix while the vech operator stacks the
columns of the lower triangular part of a matrix.
38
L(�; �) = T log
��
��k + 1
2�
��� T log
��
�1
2�
��� Tk
2log
�1� 2��
�� Tk
2log(�)
�TXt=2
�1
2log jHtj+
��k + 1
2�
�log
�1 +
�
1� 2��0tR
�1t �t
��(1.7)
where � is the inverse of the degrees of freedom as a measure of tail thickness. We assume
0 < � < 0:5 in order to have existence of the second order moments.
As Newey and Steigerwald (1997) pointed out, one concern when maximizing the
log-likelihood function based on a Student-t distribution is that estimators can be in-
consistent if the data does not follow a Student-t distribution. However, this will not be
the case as long as both the true and assumed distributions are symmetric.
Under Gaussianity assumption, one-step estimators of the parameters, �, obtained
by maximizing the corresponding likelihood function in (1.6), are consistent and asymp-
totically normal. In particular,
pn(b�n � �0) �A N(0; A�10 B0A
�10 )
where A0 is the negative expectation of the Hessian matrix evaluated at the true pa-
rameter vector �0 and B0 is the expectation of the outer product of the score vector
evaluated at �0 obtained from the likelihood function in (1.6).
If data is assumed to follow a Student-t distribution, one-step estimators of the
parameters, �, computed by maximizing the likelihood function in (1.7), are consistent
and asymptotically normal; see Fiorentini et al: (2003). It is important to note that
if the true distribution of the data is Student-t, Maximum Likelihood (ML) estimators
(in this case, one-step estimators using (1.7)) are more e¢ cient than Quasi-Maximum
Likelihood (QML) estimators obtained from maximizing the likelihood function under
the normality assumption given in (1.6).
1.3.1.2 Two-steps Estimation
It is possible to estimate the parameters of the model, � = (�0; �0; 0)0 in two steps
following Engle (2002) and Engle and Sheppard (2001). They proposed to use two-steps
when estimating the parameters of the DCC-GARCH model. The idea is to separate the
estimation of the correlation parameters, , from the mean and variance parameters, �
and � respectively.
In the �rst step, the mean and variance parameters, � and �, are estimated by
maximizing the Gaussian log-likelihood function in (1.6) in which the correlation matrix
39
Rt is replaced by the identity matrix. Therefore, in the �rst step, the function to be
maximized is the following:
L1(�; �) = �Tk
2log(2�)�
TXt=2
log jDtj �1
2
TXt=2
� 0t�t
If volatility spillovers are not allowed, i.e. A and G in equation (1.4) are restricted
to be diagonal, the �rst step estimation is equivalent to estimating k univariate models
separately; see Engle and Sheppard (2001) for details.
In the second step, given the estimates from the �rst step, b� and b�, the correlationcoe¢ cients are estimated by maximizing the following function
L2
� jb�; b�� = �1
2
TXt=2
�log jRtj+ b� 0tR�1
t b�t� (1.8)
where b�t are the standardized residuals obtained in the �rst step.Bollerslev (1990) shows that when the correlations are constant over time, i.e. in the
CCC-GARCH model, the correlation coe¢ cients estimator obtained in the second step
is equal to the sample correlation matrix of the standardized residuals given in (1.5).
If data is assumed to follow a normal distribution, two-steps estimators are also con-
sistent. Furthermore, Engle and Sheppard (2001) give conditions for the DCC-GARCH
model under which two-steps estimators are also asymptotically normal.
Next, we also consider two-steps estimation using the log-likelihood function based
on the Student-t distribution. Accordingly, in the �rst step the function to be maximized
is the multivariate Student-t log-likelihood function in (1.7) where the correlation matrix
Rt has been replaced by Ik. That is
L1(�; �; �) = T log
��
��k + 1
2�
��� T log
��
�1
2�
��� Tk
2log
�1� 2��
�� Tk
2log(�)
�TXt=2
�log jDtj+
��k + 1
2�
�log
�1 +
�
1� 2��0t�t
��Similar to the case of Gaussian innovations, when no volatility spillovers are con-
sidered, we employ univariate estimation for each series while when there are volatility
spillovers, we solve the multivariate problem. In the second step the correlation coe¢ -
cients are estimated by maximizing the following function
L2
� ; �jb�; b�� = � TX
t=2
�1
2log jRtj+
��k + 1
2�
�log
�1 +
�
1� 2�b� 0tR�1t b�t�� (1.9)
where b�t are the standardized residuals obtained in the �rst step.40
1.3.1.3 Three-steps Estimation
An alternative procedure that we will analyze in this paper is the estimation of � =
(�0; �0; 0)0 in three steps. In the �rst step, the parameters of the mean equation, �,
are estimated assuming constant variance, i.e. hit = hi 8 t, and assuming that the
correlation matrix Rt is equal to the identity matrix for all t. Therefore, the function to
be maximized is the following
L1(�; hi) = �Tk
2log(2�)�
TXt=2
log jDj � 12
TXt=2
� 0t�t
where D = diag(h1=21 ; h
1=22 ; :::; h
1=2k ) contains the conditional standard deviations. This is
equivalent to OLS estimation for the univariate mean equations, given that the variance-
covariance matrix is block diagonal.
In the second step, the parameters of the variance equation, �, are estimated given
the estimates of the parameters of the mean equation, b�, and substituting the correlationmatrix Rt by Ik. This leads to the maximization of the following function:
L2
��jb�� = �Tk
2log(2�)�
TXt=2
log jDtj �1
2
TXt=2
~� 0t~�t
where ~�t = D�1t b"t and b"t are the residuals obtained in the �rst step. After obtainingb� and b� from the two previous steps, in the last step, the correlation coe¢ cients are
estimated by maximizing the following function
L3
� jb�; b�� = �1
2
TXt=2
�log jRtj+ b� 0tR�1
t b�t� (1.10)
where b�t are the standardized residuals obtained from the second step. When the cor-
relations are constant over time, the correlation coe¢ cients estimator obtained in the
third step is, as in the two steps estimation procedure, equal to the sample correlation
matrix of the standardized residuals given in (1.5).
Under the Gaussianity assumption, three-step estimators are also consistent and their
asymptotic distribution is very similar to that of the two-step estimators; see Engle and
Sheppard (2001).
When using the log-likelihood function based on the Student-t distribution, the three
steps estimation is performed in a similar manner. In the �rst step, the mean parameters,
�, are estimated along with the inverse of the degrees of freedom assuming homoskedastic
innovations, i.e. hit = hi 8 t. The function to be maximized in the �rst step is thefollowing
L1(�; �; hi) = T log
��
��k + 1
2�
��� T log
��
�1
2�
��� Tk
2log
�1� 2��
�� Tk
2log(�)
41
�TXt=2
�log jDj+
��k + 1
2�
�log
�1 +
�
1� 2��0t�t
��In the second step, the variance parameters, �, and the inverse of the degrees of freedom,
�, are estimated conditional on the mean parameter estimates, b�. The function to bemaximized is the following
L2
��; �jb�� = T log
��
��k + 1
2�
��� T log
��
�1
2�
��� Tk
2log
�1� 2��
�� Tk
2log(�)
�TXt=2
�log jDtj+
��k + 1
2�
�log
�1 +
�
1� 2� ~�0t~�t
��Finally, in the third step, the correlation coe¢ cients and the inverse of the degrees of
freedom are estimated by maximizing the following function
L3
� ; �jb�; b�� = � TX
t=2
�1
2log jRtj+
��k + 1
2�
�log
�1 +
�
1� 2�b� 0tR�1t b�t�� (1.11)
where b�t are the standardized residuals obtained in the second step.1.3.2 Vector Autoregressive RSDC-GARCH model
The mean, variance and correlation parameters � = (�0; �0; 0)0 when Rt in equation
(1.3) is speci�ed by the RSDC-GARCH model can also be estimated in multiple steps.
Let us denote by t�1 all previous information up to t�1 and let f(�) be the likelihoodfunction obtained under the assumption of either a Gaussian or a Student-t distribution.
The one-step estimator of � would be obtained by maximizing the following log-likelihood
function:
L(�) =TXt=2
log f(Ytjt�1) (1.12)
where
f (Ytjt�1) = f (YtjSt = L;t�1)�Pr (St = Ljt�1)+f (YtjSt = H;t�1)�Pr (St = Hjt�1)
The function f (YtjSt;t�1) is the likelihood function of Yt conditional on the state St,that can be L or H, and all previous information. The function f (Ytjt�1) is thelikelihood when the state is marginalized out. On the other hand, Pr (Stjt�1) denotesthe probability of being in a certain state, St, conditional on previous information. This
probability can be computed using Hamilton �lter (Hamilton, 1994, Chapter 22). In the
case of a model with only two states, as the one analyzed in this section, Pr(Stjt�1) isgiven by:
42
Pr�St= Ljt�1
�=(1� �H;H)+ (�L;L+�H;H�1)�
� f(Y t�1jSt�1=L;t�2)�Pr (St�1=Ljt�2)f(Y t�1jSt�1=L;t�2)�Pr (St�1=Ljt�2)+f(Y t�1jSt�1=H;t�2)�(1�Pr(St�1=Ljt�2))
and consequently, Pr (St = Hjt�1) = 1 � Pr (St = Ljt�1). The long run probabilitiesfor each state are used as initial conditions for the iterative process.
Alternatively, the estimation of � = (�0; �0; 0)0 can be done in two steps. In the
�rst step, estimates of the mean and variance parameters are obtained from maximizing
the function in (1.12) where the correlation matrix Rt is substituted by the identity
matrix. In the second step, the estimation of the correlation parameters will be done by
maximizing the log-likelihood function taking the mean and variance parameter estimates
from previous step as given.
Another alternative is the estimation of � = (�0; �0; 0)0 in three steps. In the �rst
step, estimates of the mean parameters are obtained from maximizing the function in
(1.12) where the variance and correlation matrix Rt are assumed to be constant. In
the second step, variance parameters are estimated conditional on the mean parameters
obtained in the previous step, and �nally, the estimation of the correlation parameters
will be done by maximizing the log-likelihood function taking the mean and variance
parameter estimates from the two previous steps as given.
Pelletier (2006) estimates a RSDC-GARCH model by using data on four exchange
rate series. After demeaning the data, the correlation parameters are separately es-
timated from the variance parameters. This corresponds to what we have called the
three-steps estimation procedure without paying much attention to the mean parame-
ters or a two-steps estimation method for a zero mean series.
Finally, the asymptotic properties of the one-step and multiple steps estimators of the
RSDC-GARCH model under the Gaussianity assumption are similar and can be found
in Pelletier (2006).
A summary of the well-known theoretical results about ML estimation is shown in
the following table
Distribution EstimatorTrue Assumed One-step Two-steps Three-steps
Gaussian Gaussian Consistent Consistent Consistent
Student-t Student-t Consistent . .
Student-t Gaussian Consistent Consistent Consistent
Gaussian Student-t Consistent . .
43
In the next section we will con�rm the previous theoretical results in �nite samples
and study the cases for which no theory is provided, more speci�cally, what the be-
havior of multiple steps estimators is when a Student-t distribution is assumed for the
innovations.
1.4 Monte Carlo Experiments
In this section we analyze the �nite sample performance of one-step, two-steps and
three-steps estimators when they are used to estimate the parameters of �rst order Vec-
tor Autoregressive CCC, ECCC, DCC, cDCC and RSDC-GARCH models. To compare
di¤erent estimators, true parameter values are reported together with the Monte Carlo
mean and standard deviation of the parameter estimates. In addition, kernel density
estimates of di¤erent estimators of each parameter are plotted to compare the perfor-
mance of multiple steps estimators for each sample size. Since the main interest of
practitioners in this area is not only the estimation of the parameters but more impor-
tantly, the estimation of the underlying conditional variances and covariances, we will
also look at the estimates of volatilities and correlations to compare di¤erent estima-
tors. For RSDC-GARCH models the correlations are driven by an unobservable Markov
chain and therefore, estimates of the correlation parameters will be analyzed instead of
correlation estimates.
We have carried out Monte Carlo experiments in which 1000 time series vectors of
dimension 2 or 3 for sample sizes T = 200; 500; 1000 and 5000 are generated according to
the relevant model and distribution function for the innovations. Then, the parameters
of the model are estimated using one-step, two-steps and three-steps estimators assuming
either a Gaussian or a Student-t distribution for the errors. All simulations are performed
by MATLAB computer language.
Next, we describe in detail the four di¤erent experiments we have carried out. In
the �rst one, we simulate time series vectors following the �ve vector autoregressive
multivariate GARCH models considered assuming �rst a Gaussian distribution for the
innovations and then, a Student-t distribution. Parameters, volatilities and correlations
are then estimated assuming the true data generating process and di¤erences between
one-step and multiple steps estimators are analyzed. In a second experiment we study
how robust the results obtained in the previous experiment are to the error distribu-
tion. With this objective, we simulate data from the �ve models considered assuming
a Gaussian distribution for the innovations and estimate the true model under the as-
sumption that errors follow a Student-t distribution. In addition, time series vectors are
generated using a Student-t distribution for the errors and then, true models are esti-
44
mated under the Gaussianity assumption. In a third experiment we analyze how good
or bad volatilities and correlations generated from a given model can be estimated using
a di¤erent model. Finally, in the fourth and last experiment we use a skewed Student-t
distribution to generate the data and estimate the true model under the assumption that
errors follow a symmetric distribution, Gaussian or Student-t.
1.4.1 Innovations Distributed as a Gaussian or Student-t
We start by considering the case in which data is generated and estimated assuming
a normal distribution. Let us consider a bivariate model given by equations (1.1) to
(1.3) with a diagonal matrix � and Rt = R as given by the CCC-GARCH model. The
unconditional mean and variance are �xed to 1. The mean and variance persistences
are set to be di¤erent from each other but quite high. Therefore, in this basic bivariate
model, we have 11 parameters to estimate. The true parameter values as well as Monte
Carlo means and standard deviations of one-step and multiple steps estimators are given
in Table 1.1. Two main patterns, as expected for consistent estimators, emerge from this
table. First, the di¤erences between the Monte Carlo means and true parameter values
go to zero as the sample size increases. Second, the Monte Carlo standard deviations of
the three estimators considered decrease as the sample size increases. It is remarkable the
similarities of the Monte Carlo means and standard deviations of the three estimators. In
general, it seems that the one-step estimator provides estimates with Monte Carlo means
slightly closer to the parameter values and Monte Carlo standard deviations slightly
smaller than the ones obtained for multiple-steps. However, the di¤erences among the
three estimators are practically negligible. On the other hand, we cannot conclude that
in �nite samples, multiple steps estimators over/under estimate the parameters in a
systematic manner. In order to graphically illustrate the distribution, in �nite samples,
of the di¤erent estimators, Figure 1.1 plots kernel density estimates obtained from one-
step, two-steps and three-steps estimators for the parameter values considered in Table
1.1 and sample size T = 500. As the �gure shows, the three estimators give very similar
results, even for relatively small sample sizes.
In order to check the robustness of the results, we consider di¤erent scenarios by
changing the parameter values in Table 1.1 and repeat the Monte Carlo experiment.
Table 1.2 contains the new parameter values and experiments considered. First, we
consider the case in which the unconditional variance of one of the series is more than
six times the other (Experiment 2). In addition, we repeat the experiment with the
unconditional mean of one series being larger than the other (Experiment 3). We also
consider the case in which the coe¢ cients of the �rst variance equation are changed
45
(Experiment 4). The other case we analyze is when interactions among the series are
allowed (Experiment 5). Finally, we consider a trivariate model (Experiment 6). The
results obtained from all these experiments can be summarized in tables and graphs
similar to Table 1.1 and Figure 1.1. All the results are similar to the ones discussed
before and summarized in Table 1.1 and they are not included in the paper to save space
but they are available from the authors upon request.
Since, as mentioned before, the main interest of practitioners in this area is not only
the estimation of the parameters but more importantly, the estimation of the underlying
conditional variances and covariances, we have also calculated the estimated volatilities
and correlations obtained from one-step, two-steps and three-steps estimators. For a
sample size T , let us denote by bhsi;t the estimated volatilities of series i at time t obtainedfrom estimator s (one-step, two-steps or three-steps) and denote by hi;t the true volatility
of series i at time t. Then, the di¤erence between the estimated and the true volatility
of series i could be summarized for each estimator s by
�bhsi = 1
T
TXt=1
�bhsi;t � hi;t
�(1.13)
Similarly, the di¤erence between the estimated and the true correlation of series i and j
could be summarized for each estimator s by
�bpsij = 1
T
TXt=1
�bpsij;t � pij;t�
(1.14)
Figure 1.2 plots kernel density estimates of the di¤erences between the estimated and
the true volatilities and correlations measured as in (1.13) and (1.14) for a VAR(1)-CCC-
GARCH(1,1) model with parameter values as in Experiment 1 (see Table 1.1) and sample
sizes T = 200, T = 500 and T = 1000. As the graph illustrates, one-step, two-steps and
three-steps estimators provide very similar estimated volatilities and correlations. As the
sample size increases, di¤erences between estimated and true volatilities and correlations
are becoming closer to zero. Alternatively, we have also computed the relative deviations
of the estimated volatilities and correlations from their true values, i.e.bhsi;t�hi;thi;t
;bpsij;t�pij;tpij;t
and the corresponding plots are very similar to the ones in Figure 1.2.
We have repeated the Monte Carlo experiments simulating the data from di¤erent
models. Kernel density estimates of the di¤erences between the estimated and the true
volatilities and correlations in VAR(1)-DCC, cDCC, ECCC and RSDC-GARCH(1,1)
models were computed. The parameter values used in this case for the mean equation
(1.1), i.e. �;� are the same as the ones in Table 1.1. The variance parameters in equation
46
(1.4) are also the same as the ones in Table 1.1 for VAR(1)-DCC, cDCC and RSDC-
GARCH(1,1) models. For the VAR(1)-ECCC-GARCH(1,1) model they are !1 = 0:2,
!2 = 0:3, �11 = 0:25, �12 = 0:05, �21 = 0:10, �22 = 0:20, 11 = 0:50, 12 = 0:10,
21 = 0:05 and 22 = 0:40. Finally, the correlation parameter is the same as the one
in Table 1.1 for the VAR(1)-ECCC-GARCH(1,1) model. Other correlation parameters
are �Q12 = 0:20, �1 = 0:04 and �2 = 0:94 for the VAR(1)-DCC and cDCC-GARCH(1,1)
models, and �LL = 0:80, �HH = 0:90, RL12 = 0:20 and RH12 = 0:80 for the VAR(1)-RSDC-
GARCH(1,1) model. Since the graphs are very similar to Figure 1.2 they are not included
in the paper. Consequently, our results suggest that under normal innovations, using
multiple step estimators is a reasonable strategy to estimate volatilities and correlations
in all the models considered. This �nding supports, for �nite samples, the theoretical
asymptotic results summarized in Section 1.3.
Next, we consider the case in which data is generated and estimated assuming a
Student-t distribution and we repeat the simulations for all the models. The num-
ber of degrees of freedom used in the simulations is 1�= 5. For DCC-GARCH and
cDCC-GARCH models the results are similar to the ones obtained under the normal
assumption. Figure 1.3 contains, as an example, kernel density estimates of the dif-
ferences between the estimated and the true volatilities and correlations in a VAR(1)-
DCC-GARCH(1,1) model. As we can see, one-step, two-steps and three-steps estimators
provide volatilities and correlations estimates which are very close to each other. These
�ndings are in line with the results in Bauwens and Laurent (2005) and Jondeau and
Rockinger (2005) who show that, for the DCC-GARCH model, estimating mean and
variance parameters separately from the correlation parameters provides similar out-
comes to one-step estimation. In the case of the cDCC-GARCH model, results are very
similar and the graphs are not included to save space.
However, for three of the models considered, namely the VAR(1)-CCC-GARCH(1,1),
VAR(1)-ECCC-GARCH(1,1) and VAR(1)-RSDC-GARCH(1,1) models, important dif-
ferences appear when estimating the correlations (or correlation parameters and tran-
sition probabilities for the RSDC-GARCH model) with di¤erent estimators. In this
case, one-step estimator provides the best estimates. Figure 1.4 plots kernel density
estimates of the di¤erences between the estimated and the true volatilities and corre-
lations in the VAR(1)-CCC-GARCH(1,1) model. Volatilities and correlations seem to
be underestimated when using multiple steps estimators. The �gure corresponding to
the VAR(1)-ECCC-GARCH model is very similar to Figure 1.4 and it is not included in
the paper. For the RSDC-GARCH model, Figure 1.5 contains kernel density estimates
of the di¤erences between the estimated and the true volatilities and of the correlation
parameters and the transition probabilities, instead of di¤erences from estimated to true
47
correlations. As we can see, estimates obtained with multiple steps estimators seem to be
far from the ones obtained with the one-step estimator. Therefore, our results suggest
that when innovations are distributed as a Student-t, using multiple steps estimators
under the correct error distribution might not be a good idea.
1.4.2 Robustness to Error Distribution
We are also interested in analyzing how robust the di¤erent models and estimators are
to the distribution of innovations. In that sense, we have carried out an experiment
which consists of generating data from the models considered with errors following a
Gaussian distribution and estimating the true model assuming a Student-t distribution
for the innovations. In another experiment, we simulate data in which innovations follow
a Student-t distribution and estimate the true model assuming Gaussian errors.
Figure 1.6 contains kernel density estimates of the di¤erences between the estimated
and the true volatilities and correlations in a VAR(1)-ECCC-GARCH(1,1) model when
the data is generated using a Student-t distribution for the errors and estimated assuming
Gaussian errors. Di¤erences between one-step and multiple steps seem to be, again,
negligible. Compared to the case in which the true and assumed error distributions are
both normal, the estimated densities in Figure 1.6 have fatter tails. Finally, the results
illustrate how the density estimates of the di¤erences between the estimated and the
true volatilities and correlations tend to zero as the sample size increases. Similar �gures
are obtained for the other four models.
When we simulate the data with Gaussian errors and estimate the model under the
Student-t distribution assumption, for the models considered2, i.e. VAR(1)-DCC and
cDCC-GARCH(1,1) models, �gures look very similar to the case when the true and the
assumed distribution are both normal. Figure 1.7 shows the results for the VAR(1)-DCC-
GARCH(1,1) model in this case. This similarity makes sense since Student-t distribution
has an extra parameter, namely the degrees of freedom, such that this distribution could
approximate Gaussian distribution when this parameter is su¢ ciently large. In fact, in
the experiments for this last case, we obtained very large estimates for the degrees of
freedom of the Student-t distribution.2Models VAR(1)-CCC, ECCC and RSDC-GARCH(1,1) have been excluded since, as we have pre-
viously seen, multiple steps estimators do not perform well when the estimation is done under the
assumption of Student-t innovations.
48
1.4.3 Robustness to Model
The next question we address is how bad (or well) volatilities and correlations can be
estimated when the model is misspeci�ed. We analyze the di¤erences between true
conditional volatilities and correlations and the estimated ones when the model used to
generate the data is di¤erent from the estimated model. To perform this experiment,
we take the parameter values from real data. We have considered daily returns of three
European stock market indices, BEL-20 (Brussels), DAX (Frankfurt) and FTSE-100
(London) for the period January 8, 2002 - April 30, 2009. The table below contains
some descriptive statistics of the returns series, computed as 100� log�
ptpt�1
�, of sample
size 1774.
Mean SD Skewness Kurtosis
BEL-20 -0.02 1.45 -0.05 9.12
DAX -0.01 1.74 0.15 7.80
FTSE-100 -0.01 1.41 -0.03 10.30
Using the returns series, we estimate all the �ve models considered, i.e. VAR(1)- CCC,
ECCC, DCC, cDCC and RSDC-GARCH models with no mean transmissions assuming
Gaussian errors. The results are given in Table 1.3 in which series 1, 2 and 3 correspond
to BEL-20, DAX and FTSE-100, respectively.
As we can see in the table, three-steps estimates of the mean parameters are the same,
as expected, since the mean equation is the same for all the models. Two-steps mean
parameter estimates are also very similar, with the exception of the ECCC-GARCH
model, since the variance equation is the same across the other 4 models. Correlation
estimates for the CCC and ECCC-GARCH models are also very similar. The correlation
parameter estimates of the dynamic correlation models are signi�cantly di¤erent from
zero, suggesting that correlations are not constant during this period. When looking at
the other parameters, as expected, the di¤erences between one-step, two-steps and three-
steps estimates are not very large. Figures 8 and 9 plot the volatilities and correlations
estimates respectively. We can see that estimates obtained using di¤erent estimators
are very similar. The graphs containing the correlation estimates obtained from DCC
and cDCC models show that the correlation between the returns of these markets in the
period analyzed has been changing over time.
For the Monte Carlo experiments, we take the one-step estimates obtained in this
empirical exercise as the true parameter values to generate the data sets. Given that
there are �ve models, it adds up to 25 di¤erent experiments. For each model, we gen-
erate 1000 trivariate time series vectors of sample size 1000 and given each of the time
series vectors, we estimate the �ve models considered. We perform the experiments as-
49
suming a Gaussian error distribution for generating the data and also for estimating the
parameters.
The results are reported in Table 1.4, in which the models used to generate the
data appear in the �rst column and the estimated models are in the second row. For
each series, each replication and at each time t, the relative di¤erence between estimated
volatility (correlation) and true volatility (correlation) is calculated and then the average
is computed across the number of series k, replicationsR and sample size T . For example,
for the volatilities, the relative di¤erence between the estimated and the true ones is given
by the ratio
ratiotrueh;est =1
TRk
TXt=1
RXr=1
kXi=1
bhri;t � hi;t
hi;t
!(1.15)
where in our case, k = 3, R = 1000 and T = 1000. The ratios corresponding to
the one-step estimation of a model that is correctly speci�ed is set to be equal to 0.
Therefore, the ratios reported in Table 1.4 are relative ratios and they should be read
as the performance of the corresponding estimator in a certain model when estimating
the volatility (correlation), relative to the one-step estimator in the correctly speci�ed
model. The results are reported in three parts: volatilities, correlations and covariances.
In general, we can see that the ratios are all very close to zero, indicating that, on average,
volatilities and correlations are relatively well estimated even when using a misspeci�ed
model.
More speci�cally, looking at the results for the volatilities, we can see that the largest
ratio is 0:0165 and it appears when the true volatilities are generated by the VAR(1)-
cDCC-GARCH model and estimated by the VAR(1)-ECCC-GARCH in one step. Other
large ratios correspond to three-steps estimators of all the models considered when the
data have been generated by the VAR(1)-ECCC-GARCH model. For example, when the
true volatilities are generated by the VAR(1)-ECCC-GARCH model and estimated by
the VAR(1)-CCC-GARCH in three steps, the ratio is 0:0137, when they are estimated
by the VAR(1)-DCC-GARCH, the ratio is 0:0154 and when using the VAR(1)-cDCC-
GARCH and the VAR(1)-RSDC-GARCH models to estimate the volatilities, the ratio
is 0:0141 and 0:0136 respectively. The reason could be that with the exception of the
correlation structure, all the models considered are nested in the VAR(1)-ECCC-GARCH
model, being this one more general and therefore, the rest of the models have problems
in explaining the volatilities generated by the VAR(1)-ECCC-GARCH model. On the
other hand, the true volatilities generated by the VAR(1)-CCC-GARCH model can be
well estimated by the other models since CCC-GARCH is nested within all of them.
When looking at the results for the correlations, we can see that the largest ratio
50
is 0:0071 and it appears when the true correlations are generated by the VAR(1)-DCC-
GARCH model and estimated by the VAR(1)-ECCC-GARCH in one step. As expected,
when the correlations are generated by a dynamic correlation model, their estimation is
better when assuming another dynamic correlation model than when a constant corre-
lation model is used. Also expected is the fact that the VAR(1)-ECCC-GARCH model
produces better estimates of the correlations generated by the VAR(1)-CCC-GARCH
model than the estimates produced by the VAR(1)-CCC-GARCH model when estimat-
ing the correlations generated by the VAR(1)-ECCC-GARCH model. The reason could
be that the VAR(1)-CCC-GARCH model can not capture the volatility spillovers which
indirectly can a¤ect correlations.
In general terms, when volatilities and correlations which have been generated by a
particular model are estimated by another model, their estimation seem to get worse as
the number of steps used in the estimation increase. On the other hand, the average
ratios do not deviate from zero more than 2 % in most of the cases. An interpretation of
this result could be that, on average, multiple steps estimates of volatilities (correlations)
deviate from the corresponding true volatilities (correlations) at most 2 % more than
the amount that one-step estimates of the correctly speci�ed model do.
1.4.4 Innovations Distributed as a Skewed Student-t
In this section, we analyze the case in which innovations follow a skewed Student-t
distribution. For this purpose, we generate random vectors from a skewed multivari-
ate Student-t distribution following Bauwens and Laurent (2005). At each time t; a k
dimensional random vector ��t is given by:
��t = �(�)jxtjjxtj = (jx1tj; jx2tj; :::; jxktj)0
where xt follows a multivariate Student-t distribution with zero mean and unit vari-
ance and �(�) is a k � k diagonal matrix such that:
�(�) = ��� (Ik � �)��1
� = diag(�)
� = (�1; �2; :::; �k); with �i > 0
� = diag(� 1; � 2; :::; � k); with � i 2 f0; 1g
� i v Ber
��2i
1 + �2i
�
51
where Ber�
�2i1+�2i
�is a Bernoulli distribution with probability of success �2i
1+�2iand the
elements of � are mutually independent. Given that in the GARCH set up, the elements
of �t are zero mean random numbers with unit variance, ��t should be standardized such
that �it =��it�mi
siwhere:
mi =��v�12
�pv � 2
p���v2
� ��i �
1
�i
�s2i =
��2i �
1
�2i� 1��m2
i
We �rst generate bivariate series with skewness parameters �1 = �2 = exp(0:4) for
both series, which implies a skewness of 1:5. Later we take �1 = exp(0:4) and �2 =
exp(�0:7) (implying a skewness of �2 for the second series) to see how the results
change. Notice that when �1 = �2 = 1, we have a symmetric multivariate Student-t
distribution.
For the Monte Carlo experiments we use the estimates reported in Table 1.53 as the
true parameter values. We generate 1000 bivariate time series vectors of sample size
equal to 1000. Innovations are generated from a skewed Student-t distribution with
skewness 1:5 for both series or with skewness f1:5;�2g for the �rst and second seriesrespectively and with degrees of freedom 5. Then we estimate the true model assuming
Gaussian or Student-t errors but ignoring skewness.
Figure 1.10 plots the results of the Monte Carlo experiment when data has been gen-
erated using a positively skewed Student-t distribution with the same skewness for both
series and estimated assuming Gaussian innovations. In this �gure, the rows correspond
to a di¤erent model and the columns represent the kernel densities estimates of the rel-
ative deviations of estimated volatilities and correlations from the true ones calculated
respectively as:4
�bhsi = 1
T
TXt=1
(bhsi;t � hi;t
hi;t
)(1.16)
�bpsi = 1
T
TXt=1
�bpsi;t � pi;t
pi;t
�(1.17)
As we can see, for all the models, the kernel densities of the relative deviations of
one-step and multiple steps estimates of volatilities (correlations) from the true ones
3Table 5 contains parameter estimates for the 5 models considered using daily returns of the BEL-20
and DAX stock market indices under the assumption that innovations are distributed as a Student-t.4Relative deviations are prefered to absolute ones, although conclusions do not change if absolute
deviations are plotted.
52
follow each other closely. It seems that the large positive skewness assumed in the data
generating process results in overestimation of the conditional correlations by around
2-3 % in each model while the conditional volatility estimates don�t seem to be a¤ected
much.
Figure 1.11 plots the same estimates as Figure 1.10 but now the estimation has been
done assuming a Student-t distribution for the innovations. Similar conclusions can be
made about the one step correlation estimates for all models. We notice that in the
CCC and ECCC-GARCH models, the di¤erences between one step and multiple steps
estimates of the correlations are very large.
On the other hand, when the series have di¤erent skewness and the estimation is
performed assuming Gaussian errors, volatilities and correlations seem to be underesti-
mated in all the �ve models considered. The �gures corresponding to di¤erent skewness
are not included in the paper to save space. One-step correlation estimates seem to be
slightly less a¤ected by the skewness than the multiple step estimates. As well when
the estimation is based on Student-t errors, the one-step estimators underestimate the
volatilities and correlations. In general, one-step estimators are less a¤ected by the
skewness than multiple steps estimators, except for the volatility estimates of ECCC-
GARCH model. In the case of DCC and cDCC-GARCH models, the multiple steps
estimates deviate slightly from the one step estimates. It should be noted that one of
the series have higher skewness when � = fexp(0:4); exp(�0:7)g compared to the casewhen � = fexp(0:4); exp(0:4)g and this could be the reason behind the underestimationof volatilities and correlations with both Gaussian and Student-t errors.
Newey and Steigerwald (1997) suggest that when the data is not symmetrically dis-
tributed, the one-step QML method based on Student-t errors do not produce consistent
estimators in general. Therefore in this case what is expected is that even though the
estimation is performed in one-step, the estimates could be far from the true values and
the di¤erences might not disappear in larger samples. In our experiments with a data
of length T = 1000, we see that one-step QML estimators based on Student-t errors are
over/underestimating the volatilities and correlations. We would expect that this result
holds for larger datasets produced with the same parameter values and skewness.
For the RSDC-GARCH model, multiple steps estimators of conditional volatilities
behave similar to the one-step estimators as illustrated in Figure 1.5 and this does not
seem to depend on the skewness. In this model, the conditional correlations follow
an unobserved Markov Chain, therefore instead of reporting correlation estimates, we
report the correlation parameter estimates, RL, RH , �LL, �HH together with their true
values. Figure 1.12 plots kernel density estimates of estimated correlation parameters
when the series have the same skewness and errors are assumed to follow a Gaussian or
53
Student-t distribution. As we can see, when the estimations are based on Gaussian errors,
the one-step and multiple steps estimators of the correlation parameters are behaving
similarly when the series have the same skewness. Although the corresponding �gure is
not included in the paper, when the skewness of both series is di¤erent, the multiple steps
estimates of RL and �LL deviate slightly from the one step estimates. When Student-t
errors are used in the estimation, the di¤erences between the behavior of one-step and
multiple steps estimators become more apparent.
Finally, when the data generating process is symmetric and the estimation is based
on Gaussian errors, the kernel density estimates of relative di¤erences between one-step
and multiple steps estimates of the volatilities and correlations from the true values are
very close to each other for all the models as was illustrated in Figure 1.6. Also when
the estimation is based on Student-t errors, the multiple steps correlation estimates of
CCC and ECCC-GARCH models are far from the true ones as was shown in Figure
1.4. The multiple steps estimates of DCC and cDCC-GARCH models of volatilities and
correlations follow closely the one-step estimates and are not far from the true values as
in Figure 1.3. These results are also not reported in the paper, but are available from
the authors upon request.
To sum up, we have seen that even though the data generating process is skewed,
when the estimation is based on Gaussian errors, multiple-steps estimators could still be
preferred to one-step estimators given that their performances are very similar. Given
that the estimation based on Gaussian errors is a Quasi-maximum Likelihood estimation,
as Bollerslev and Wooldridge (1992) show, it produces consistent estimators. Therefore
our results from Section 1.4.1 and 1.4.2 still prevail in the existence of skewness. On the
other hand, as noted by Newey and Steigerwald (1997), if the data generating process is
skewed, the one-step QML estimator based on Student-t errors do not produce consistent
estimators. In conformance with this, we have found in this section that the correlations
are over-estimated in all models with one-step and also with multiple steps estimators.
Hence, when the true distribution is skewed, one should be cautious in using one-step or
multiple-steps estimators based on Student-t errors.
1.5 Conclusions
In this paper we have carried out several Monte Carlo experiments to study the perfor-
mance in �nite samples of one-step and multiple steps estimators of Vector Autoregressive
Multivariate Conditional Correlation GARCH models. Although one-step estimators
are preferable because of their theoretical properties, they are not always feasible and
therefore, estimating the parameters of a model in multiple steps could be a reasonable
54
alternative. Our results indicate that, when the distribution of the errors is Gaussian,
multiple steps estimators have a very good performance even in small samples. However,
when the estimation is based on Student-t errors, we �nd that multiple steps estimators
do not always perform well even when the data follows a Student-t distribution.
Our results also show that if the true error distribution is Student-t but estima-
tion is based on the Gaussian distribution, kernel density estimates of the estimates of
volatility and correlation obtained from one-step and multiple steps estimators are quite
similar. Analogously, if the true error distribution is Gaussian but estimation is based
on the Student-t distribution, we obtain the same results as when the true and assumed
distribution is a Student-t.
We also analyze the robustness of our results to the misspeci�cation of the model
when the estimation is based on Gaussian errors. We �nd that, on average, volatilities
and correlations are relatively well estimated even when using a misspeci�ed model. The
multiple-steps estimates of volatilities (correlations) deviate from the true values at most
by 2 % more than what one-step estimates of the correctly speci�ed model do.
Finally, when errors are distributed as a skewed Student-t but the estimation is
performed assuming non-skewed Gaussian or Student-t errors, we �nd that kernel density
estimates of the di¤erence between one-steps and multiple steps estimates of volatilities
and correlations from their true values are very similar when the estimation is based
on a Gaussian distribution. However, this is not true when the estimation is based
on Student-t errors. In any case, when the true distribution is skewed, one should be
cautious in using one-step or multiple-steps estimators based on Student-t errors since
both are inconsistent estimators.
55
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Figure 1.1: Kernel density estimates for estimated parameters of a VAR(1)-CCC-
GARCH(1,1) model with T = 500
0 0.2 0.4 0.6 0.8 10
5
10
15µ1
0 0.2 0.4 0.6 0.8 10
5
10
15β1
0 0.2 0.4 0.6 0.8 10
5
10
15µ2
0 0.2 0.4 0.6 0.8 105
1015
ω1
0 0.2 0.4 0.6 0.8 10
5
10
15β2
0 0.2 0.4 0.6 0.8 10
5
10
15α1
0 0.2 0.4 0.6 0.8 10
5
10
15γ1
0 0.2 0.4 0.6 0.8 10
5
10
15ω2
0 0.2 0.4 0.6 0.8 10
5
10
15α2
0 0.2 0.4 0.6 0.8 10
5
10
15γ2
0 0.2 0.4 0.6 0.8 10
5
10
15α1+γ1
0 0.2 0.4 0.6 0.8 10
5
10
15α2+γ2
0 0.2 0.4 0.6 0.8 10
5
10
15ρ
1s2s3strue value
CCCnCCCn
Sample size: 500
59
Figure 1.2: Kernel density estimates of deviations from estimated to true volatility in a
VAR(1)-CCC-GARCH(1,1) model with Gaussian innovations
0.5 0 0.50
2
4
6
8
10∆ h1
s
0.5 0 0.50
2
4
6
8
10∆ h1
s
0.5 0 0.50
2
4
6
8
10∆ h2
s
0.5 0 0.50
2
4
6
8
10∆ h1
s
0.5 0 0.50
2
4
6
8
10∆ h2
s
0.5 0 0.50
2
4
6
8
10∆ h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
1s2s3szero line
Sample size:500 Sample size:1000Sample size:200
60
Figure 1.3: Kernel density estimates of deviations from estimated to true volatility in a
VAR(1)-DCC-GARCH(1,1) model with Student-t innovations
0.5 0 0.50
2
4
6
8
10∆h1
s
1s2s3szero line
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
Sample size:200 Sample size:500 Sample size:1000
61
Figure 1.4: Kernel density estimates of deviations from estimated to true volatility in a
VAR(1)-CCC-GARCH(1,1) model with Student-t innovations
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
1s2s3szero line
Sample size:200 Sample size:500 Sample size:1000
62
Figure 1.5: Kernel density estimates of deviations from estimated to true volatility, of
estimated correlation parameters and of estimated transition probabilities in a VAR(1)-
RSDC-GARCH(1,1) model with Student-t innovations
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
1 0 10
2
4
6
8
10RL
1 0 10
2
4
6
8
10RH
0 0.5 10
2
4
6
8
10πLL
0 0.5 10
2
4
6
8
10πHH
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
1 0 10
2
4
6
8
10RL
1 0 10
2
4
6
8
10RH
0 0.5 10
2
4
6
8
10πLL
0 0.5 10
2
4
6
8
10πHH
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
1 0 10
2
4
6
8
10RL
1 0 10
2
4
6
8
10RH
0 0.5 10
2
4
6
8
10πLL
0 0.5 10
2
4
6
8
10πHH
1s2s3s
Sample size:200 Sample size:500 Sample size:1000
63
Figure 1.6: Kernel density estimates of deviations from estimated to true volatility in a
VAR(1)-ECCC-GARCH(1,1) model generated with Student-t innovations and estimated
assuming Gaussian errors
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
1s2s3szero line
Sample size:200 Sample size:500 Sample size:1000
64
Figure 1.7: Kernel density estimates of deviations from estimated to true volatility in a
VAR(1)-DCC-GARCH(1,1) model generated with Gaussian innovations and estimated
assuming Student-t errors
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
0.5 0 0.50
2
4
6
8
10∆h1
s
0.5 0 0.50
2
4
6
8
10∆h2
s
0.2 0.1 0 0.1 0.20
5
10
15
20∆ps
1s2s3szero line
Sample size:200 Sample size:500 Sample size:1000
65
Figure 1.8: One-step, two-steps and three-steps estimates of the volatilities of BEL-20,
DAX and FTSE-100 observed from January 8, 2002 to April 30, 2009, asuming Gaussian
innovations.
0 500 1000 15000
10
20
30BEL20
CC
C
0 500 1000 15000
10
20
30DAX
0 500 1000 15000
10
20
30FTSE100
0 500 1000 15000
10
20
30
EC
CC
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
DC
C
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
cDC
C
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
RS
DC
0 500 1000 15000
10
20
30
0 500 1000 15000
10
20
30
1s2s3s
66
Figure 1.9: One-step, two-steps and three-steps estimates of the correlations between
BEL-20, DAX and FTSE-100 indices observed from January 8, 2002 to April 30, 2009,
asuming Gaussian innovations.
0 500 1000 15000.4
0.6
0.8
1BEL20&DAX
CC
C
0 500 1000 15000.4
0.6
0.8
1BEL20&FTSE100
0 500 1000 15000.4
0.6
0.8
1DAX&FTSE100
0 500 1000 15000.4
0.6
0.8
1
EC
CC
0 500 1000 15000.4
0.6
0.8
1
0 500 1000 15000.4
0.6
0.8
1
0 500 1000 15000.4
0.6
0.8
1
DC
C
0 500 1000 15000.4
0.6
0.8
1
0 500 1000 15000.4
0.6
0.8
1
0 500 1000 15000.4
0.6
0.8
1
cDC
C
0 500 1000 15000.4
0.6
0.8
1
0 500 1000 15000.4
0.6
0.8
1
1s2s3s
67
Figure 1.10: Kernel density estimates of deviations from estimated to true volatility and
correlation for all the models considered. The series in the data are generated using
Student-t innovations with same skewness parameter and estimated assuming Gaussian
innovations.
0.5 0 0.50
5
10∆h1
s
CC
C
0.5 0 0.50
5
10∆h2
s1s2s3szero line
0.2 0.1 0 0.1 0.20
10
20∆ps
0.5 0 0.50
5
10
EC
CC
0.5 0 0.50
5
10
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
DC
C
0.5 0 0.50
5
10
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
cDC
C
0.5 0 0.50
5
10
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
RS
DC
0.5 0 0.50
5
10
68
Figure 1.11: Kernel density estimates of deviations from estimated to true volatility and
correlation for all the models considered. The series in the data are generated using
Student-t innovations with same skewness parameter and estimated assuming Student-t
innovations.
0.5 0 0.50
5
10∆h1
s
CC
C
0.5 0 0.50
5
10∆h2
s
0.2 0.1 0 0.1 0.20
10
20∆ps
0.5 0 0.50
5
10
EC
CC
0.5 0 0.50
5
10
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
DC
C
1s2s3szero line
0.5 0 0.50
5
10
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
0.5 0 0.50
5
10
cDC
C
0.2 0.1 0 0.1 0.20
10
20
0.5 0 0.50
5
10
RS
DC
0.5 0 0.50
5
10
69
Figure 1.12: Kernel density estimates of estimated correlation parameters for the RSDC-
GARCHmodel. The series in the data are generated with Student-t innovations and with
same skewness, and estimated assuming Gaussian and Student-t errors, respectively.
1 0.5 0 0.5 10
5
10
15RL
1s2s3strue value
1 0.5 0 0.5 10
5
10
15RH
0 0.5 10
5
10
15πLL
0 0.5 10
5
10
15πHH
1 0.5 0 0.5 10
5
10
15RL
1 0.5 0 0.5 10
5
10
15RH
0 0.5 10
5
10
15πLL
0 0.5 10
5
10
15πHH
Gaussian Studentt
70
Table1.1:MonteCarlomeanandstandarddeviationsofone-step,two-stepsandthree-stepsestimatorsofabivariateGaussian
VAR(1)-CCC-GARCHmodel
One-step
Two-steps
Three-steps
Parameter
Value
T=500
T=1000
T=5000
T=500
T=1000
T=5000
T=500
T=1000
T=5000
�1
0:20
0:207
(0:050)
0:204
(0:036)
0:201
(0:016)
0:207
(0:050)
0:204
(0:037)
0:201
(0:017)
0:208
(0:053)
0:204
(0:039)
0:201
(0:017)
�2
0:40
0:403
(0:060)
0:403
(0:043)
0:400
(0:017)
0:403
(0:061)
0:403
(0:044)
0:400
(0:018)
0:403
(0:062)
0:404
(0:044)
0:400
(0:018)
�1
0:80
0:793
(0:028)
0:796
(0:020)
0:799
(0:009)
0:793
(0:029)
0:796
(0:020)
0:799
(0:009)
0:792
(0:030)
0:796
(0:022)
0:799
(0:010)
�2
0:60
0:596
(0:038)
0:598
(0:026)
0:600
(0:011)
0:597
(0:039)
0:598
(0:027)
0:600
(0:012)
0:596
(0:039)
0:597
(0:027)
0:600
(0:012)
!1
0:10
0:180
(0:179)
0:124
(0:079)
0:103
(0:019)
0:182
(0:183)
0:123
(0:072)
0:103
(0:019)
0:183
(0:184)
0:124
(0:075)
0:103
(0:019)
!2
0:05
0:270
(0:308)
0:120
(0:177)
0:053
(0:015)
0:273
(0:311)
0:132
(0:198)
0:053
(0:015)
0:290
(0:339)
0:146
(0:231)
0:054
(0:031)
�1
0:10
0:108
(0:044)
0:103
(0:030)
0:099
(0:012)
0:109
(0:044)
0:103
(0:030)
0:099
(0:013)
0:106
(0:043)
0:102
(0:030)
0:099
(0:013)
�2
0:05
0:061
(0:036)
0:054
(0:023)
0:050
(0:009)
0:061
(0:037)
0:054
(0:024)
0:050
(0:009)
0:061
(0:035)
0:054
(0:023)
0:050
(0:009)
1
0:80
0:706
(0:203)
0:772
(0:096)
0:796
(0:027)
0:705
(0:206)
0:773
(0:089)
0:796
(0:027)
0:705
(0:208)
0:772
(0:093)
0:797
(0:027)
2
0:90
0:660
(0:322)
0:822
(0:192)
0:897
(0:021)
0:656
(0:325)
0:810
(0:212)
0:897
(0:021)
0:637
(0:355)
0:796
(0:243)
0:896
(0:035)
�0:20
0:199
(0:044)
0:201
(0:031)
0:200
(0:014)
0:198
(0:043)
0:199
(0:031)
0:200
(0:014)
0:198
(0:043)
0:199
(0:031)
0:200
(0:014)
71
Table 1.2: Parameter values of the VAR(1)-CCC-GARCH model for di¤erent Monte
Carlo experiments
Parameter Basic (Table 1) Experiment 2 Experiment 3 Experiment 4 Experiment 5 Experiment 6
�1 0:20 0:20 0:30 0:20 0:10 0:20
�2 0:40 0:40 0:40 0:40 0:40 0:40
�3 - - - - - 0:30
�11 0:80 0:80 0:80 0:80 0:80 0:80
�12 0:00 0:00 0:00 0:00 0:10 0:00
�21 0:00 0:00 0:00 0:00 0:10 0:00
�22 0:60 0:60 0:60 0:60 0:60 0:60
�33 - - - - - 0:70
!1 0:10 0:10 0:10 0:10 0:10 0:10
!2 0:05 1:00 0:05 0:05 0:05 0:05
!3 - - - - - 0:05
�1 0:10 0:10 0:10 0:35 0:10 0:10
�2 0:05 0:15 0:05 0:05 0:05 0:05
�3 - - - - - 0:15
1 0:80 0:80 0:80 0:55 0:80 0:80
2 0:90 0:70 0:90 0:90 0:90 0:90
3 - - - - - 0:80
�12 0:20 0:20 0:20 0:20 0:20 0:10
�13 - - - - - 0:20
�23 - - - - - 0:30
72
Table1.3:ParameterestimatesforthreerealtimeseriesunderGaussianinnovations
VAR(1)-CCC-GARCH
VAR(1)-ECCC-GARCH
VAR(1)-DCC-GARCH
VAR(1)-cDCC-GARCH
VAR(1)-RSDC-GARCH
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps
3-steps
�1
0:1017
0:0937
�0:0167
0:0997
0:0969
�0:0167
0:0982
0:0937
�0:0167
0:1008
0:0936
�0:0166
0:0851
0:0936
�0:0166
�2
0:1110
0:0843
�0:0061
0:1097
0:0707
�0:0061
0:1041
0:0843
�0:0061
0:1065
0:0843
�0:0062
0:0927
0:0843
�0:0061
�3
0:0690
0:0465
�0:0128
0:0683
0:0503
�0:0128
0:0646
0:0463
�0:0128
0:0669
0:0465
�0:0128
0:0611
0:0465
�0:0128
�1
�0:0297
�0:0055
0:0641
�0:0267
�0:0112
0:0641
�0:0277
�0:0055
0:0640
�0:0278
�0:0055
0:0641
�0:0245
�0:0055
0:0640
�2
�0:0934
�0:0549
�0:0465
�0:0903
�0:0455
�0:0465
�0:0748
�0:0549
�0:0466
�0:0738
�0:0549
�0:0465
�0:0743
�0:0549
�0:0466
�3
�0:0991
�0:1033
�0:0821
�0:0956
�0:1020
�0:0820
�0:0938
�0:1027
�0:0821
�0:0935
�0:1034
�0:0820
�0:0888
�0:1034
�0:0820
!1
0:0288
0:0218
0:0210
0:0322
0:0101
0:0170
0:0248
0:0218
0:0209
0:0240
0:0218
0:0210
0:0233
0:0218
0:0210
!2
0:0259
0:0210
0:0201
0:0245
0:0183
0:0172
0:0226
0:0210
0:0201
0:0213
0:0210
0:0201
0:0212
0:0210
0:0201
!3
0:0171
0:0102
0:0097
0:0168
0:0162
0:0000
0:0143
0:0094
0:0098
0:0131
0:0102
0:0097
0:0173
0:0102
0:0098
�11
0:0940
0:1331
0:1231
0:0772
0:0402
0:0489
0:1118
0:1331
0:1231
0:1183
0:1331
0:1231
0:0874
0:1331
0:1231
�21
0:0032
0:0328
0:0249
�31
0:0224
0:0684
0:0373
�12
0:0000
0:0000
0:0000
�22
0:0774
0:0955
0:0927
0:0794
0:0678
0:0676
0:0887
0:0955
0:0927
0:0933
0:0955
0:0927
0:0687
0:0955
0:0927
�32
0:0038
0:0000
0:0000
�13
0:0476
0:0907
0:0973
�23
0:0000
0:0000
0:0051
�33
0:0777
0:1045
0:1041
0:0599
0:0605
0:0343
0:0935
0:0938
0:1041
0:0985
0:1045
0:1041
0:0779
0:1045
0:1041
11
0:8819
0:8573
0:8673
0:8399
0:0000
0:8577
0:8705
0:8573
0:8673
0:8695
0:8573
0:8673
0:8956
0:8573
0:8673
21
0:0000
0:0000
0:0000
31
0:0000
0:2938
0:6074
12
0:0083
0:0000
0:0000
22
0:9095
0:8977
0:9011
0:9065
0:9023
0:9054
0:9005
0:8977
0:9011
0:9004
0:8977
0:9011
0:9217
0:8977
0:9011
32
0:0029
0:0000
0:0000
13
0:0000
0:9680
0:0000
23
0:0000
0:0000
0:0000
33
0:9063
0:8921
0:8936
0:8883
0:5322
0:2512
0:8948
0:9015
0:8936
0:8948
0:8921
0:8936
0:9088
0:8921
0:8936
�12
0:7911
0:7865
0:7866
0:7921
0:7853
0:7866
�L 12
0:6571
0:6674
0:6455
�H 12
0:8782
0:8802
0:8773
�13
0:7751
0:7642
0:7644
0:7770
0:7662
0:7654
�L 13
0:6286
0:6286
0:6060
�H 13
0:8695
0:8702
0:8658
�23
0:8050
0:8013
0:8016
0:8054
0:7983
0:8000
�L 23
0:6477
0:6633
0:6371
�H 23
0:9054
0:9087
0:9063
�1
0:0411
0:0459
0:0494
0:0405
0:0439
0:0449
�2
0:9215
0:9172
0:9116
0:9272
0:9226
0:9217
�LL
0:8718
0:8934
0:8681
�HH
0:9272
0:9213
0:9207
73
Table1.4:Volatility,covarianceandcorrelationratios
Simulatedmodel
Estimatedmodel
VAR(1)-CCC-GARCH
VAR(1)-ECCC-GARCH
VAR(1)-DCC-GARCH
VAR(1)-cDCC-GARCH
VAR(1)-RSDC-GARCH
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps
3-steps
1-step
2-steps3-steps
VAR(1)-CCC-GARCH
0:0000�0:0034
0:0041
0:0021�0:0025
0:0029�0:0032�0:0030
0:0049�0:0023�0:0033
0:0039�0:0023�0:0035
0:0046
VAR(1)-ECCC-GARCH
0:0095
0:0064
0:0137
0:0000�0:0001
0:0023
0:0078
0:0065
0:0154
0:0080
0:0065
0:0141
0:0089
0:0064
0:0136
VAR(1)-DCC-GARCH
Volatility
0:0083
0:0035
0:0120
0:0143
0:0042
0:0059
0:0000
0:0035
0:0104
0:0044
0:0037
0:0099
0:0050
0:0032
0:0107
VAR(1)-cDCC-GARCH
0:0014
0:0005
0:0092
0:0165
0:0015
0:0068�0:0031
0:0010
0:0080
0:0000
0:0005
0:0102
0:0024�0:0002
0:0086
VAR(1)-RSDC-GARCH
�0:0010�0:0008
0:0080
0:0067
0:0005
0:0013�0:0027�0:0009
0:0068�0:0027�0:0009
0:0068
0:0000�0:0007
0:0080
VAR(1)-CCC-GARCH
0:0000�0:0022�0:0039
0:0007�0:0027�0:0034�0:0003�0:0023�0:0042�0:0004�0:0022�0:0040
VAR(1)-ECCC-GARCH
�0:0024�0:0045�0:0062
0:0000�0:0031�0:0034�0:0026�0:0045�0:0064�0:0025�0:0044�0:0062
VAR(1)-DCC-GARCH
Correlation
0:0056
0:0022
0:0007
0:0071
0:0013
0:0010
0:0000�0:0021�0:0036
0:0019�0:0004�0:0021
VAR(1)-cDCC-GARCH
0:0033
0:0005�0:0012
0:0059�0:0003�0:0012�0:0015�0:0039�0:0052
0:0000�0:0023�0:0045
VAR(1)-RSDC-GARCH
VAR(1)-CCC-GARCH
0:0000�0:0035�0:0025�0:0003�0:0034�0:0037�0:0026�0:0039�0:0022�0:0019�0:0036�0:0017
VAR(1)-ECCC-GARCH
0:0135
0:0025
0:0035
0:0000�0:0051�0:0048
0:0108
0:0025
0:0044
0:0124
0:0029
0:0038
VAR(1)-DCC-GARCH
Covariance
0:0255�0:0003�0:0034
0:0232�0:0024�0:0015
0:0000�0:0057�0:0051
0:0066
0:0010
0:0000
VAR(1)-cDCC-GARCH
0:0198�0:0050�0:0070
0:0210�0:0063�0:0086�0:0048�0:0155�0:0132
0:0000�0:0069�0:0130
VAR(1)-RSDC-GARCH
74
Table1.5:One-stepparameterestimatesfortworealtimeseriesunderStudent-tinnovations
VAR(1)-CCC-GARCH
VAR(1)-ECCC-GARCH
VAR(1)-DCC-GARCH
VAR(1)-cDCC-GARCH
VAR(1)-RSDC-GARCH
�1
0:0936
0:0937
0:0956
0:0963
0:0920
�2
0:1090
0:1093
0:1143
0:1136
0:1100
�1
0:0087
0:0088
0:0002
0:0006
0:0030
�2
�0:0503
�0:0498
�0:0488
�0:0469
�0:0485
!1
0:0185
0:0177
0:0156
0:0149
0:0160
!2
0:0191
0:0162
0:0152
0:0141
0:0156
�11
0:0877
0:0947
0:0963
0:0978
0:0894
�21
0:0001
�12
0:0034
�22
0:0732
0:0692
0:0775
0:0790
0:0717
11
0:8987
0:8789
0:8938
0:8950
0:8998
21
0:0086
12
0:0001
22
0:9195
0:9225
0:9170
0:9181
0:9218
�12
0:7950
0:7949
�L 12
0:7298
�H 12
0:8924
�1
0:0376
0:0390
�2
0:9453
0:9465
�LL
0:9816
�HH
0:9682
75
Chapter 2
Estimation of MultivariateStochastic Volatility Models: AComparative Monte Carlo Study
2.1 Introduction
In �nancial time series literature, it is already established that the volatilities of asset
returns are changing over time. Moreover, they are likely to be serially correlated.
To illustrate this stylized fact with an example, in Figure 2.1 we present the indices
and returns (100 x log(Pt=Pt�1)) of FTSE-100 and DAX stock markets between dates
4/1/2005 and 4/11/2011. We also plot the squared returns, as a proxy of volatilities, and
a rolling window estimate of correlations, with a window of 60 days. It is observed that
the volatilities are changing over time. Moreover, the volatilities are clustered; i.e. higher
(lower) values of volatilities are followed by higher (lower) values, which implies that the
volatilities are serially correlated. To capture this kind of a dynamic volatility e¤ect, the
generalized autoregressive conditional heteroskedasticity (GARCH) models have been
proposed by Engle (1982) and Bollerslev (1986). In GARCH models the time varying
volatility is modelled as a deterministic function of squared previous day returns and
previous day volatilities; therefore in GARCH approach the volatilities are observation
driven. Currently a wide range of GARCH models are available in the literature and are
well documented in the surveys: see Bollerslev et al. (1992) for univariate and Bauwens
et al. (2006), Silvennoinen et al. (2009) for multivariate models.
An alternative approach to modelling time varying volatility is to consider it as
an unobserved component and let the logarithm of it follow an autoregressive process.
Therefore in this approach, the volatilities are parameter driven. Models of this kind
77
are named as stochastic volatility (SV) models in the literature. The SV approach is
attractive because of its similarity to the models used in �nancial theory to describe the
behavior of prices; see Hull and White (1987), Taylor (1986, 1994), and Shephard and
Andersen (2008) for the origins of SV models. Moreover it has been shown that the SV
models describe the behavior of volatilities more accurately compared to GARCHmodels
(see for example Danielsson (1994), Kim et al. (1998), and Carnero et al. (2004)). Given
the way the SV models are set up, their statistical properties are easy to derive from
the process that the volatilities follow. However, although statistically more attractive
than GARCH models, SV models have the disadvantage in terms of estimation because
their exact likelihoods are di¢ cult to evaluate. The following survey papers are available
about the univariate and multivariate SV models and estimation methods: Broto and
Ruiz (2004), Asai et al. (2006), Chib et al. (2009), Ghysels et al. (1996), Yu and Meyer
(2004), Maasoumi and McAleer (2006).
Several methods have been proposed for estimating SV models. A relatively easy
approach is the quasi-maximum likelihood estimation (QML) proposed independently
by Nelson (1988) and Harvey et al. (1994). In this approach, the log-squared returns are
modelled as a linear state space form where the transformed innovations are assumed
to follow a Gaussian distribution although in fact the true distribution is based on ln�21(see Sandman and Koopman (1998) for the univariate and Asai and McAleer (2006) for
the multivariate case). Ruiz (1994) showed that the QML estimators are consistent and
asymptotically normal. However due to the Gaussianity assumption, QML approach
is an estimation based on approximations and therefore, as noted by several papers
as Jacquier et al. (1994), Breidt and Carriquiry (1996) and Sandmann and Koopman
(1998), QML estimator is ine¢ cient.
The evaluation of exact likelihood requires high dimensional integration which could
be based on evaluating these integrals with simulation methods and then maximizing
the resulting likelihood function. This class of estimation approaches include the acceler-
ated importance sampling (AGIS) approach developed in Danielsson and Richard (1993)
and e¢ cient importance sampling (EIS) approach proposed by Liesenfeld and Richard
(2003, 2006), and the Monte Carlo likelihood (MCL) approach proposed by Sandman
and Koopman (1998). Di¤erent from the QML estimation, the MCL method of Sandman
and Koopman (1998) used log-squared transformation of returns taking into account the
true distribution of the errors and therefore modelling the log-squared returns via a lin-
ear non-Gaussian state space model. A review of these importance sampling methods
could be found again in Asai et al. (2006).
The MCL method considered in this paper is the one proposed by Jungbacker and
Koopman (2006) that extended the theoretical results of Shephard and Pitt (1997),
78
Durbin and Koopman (1997), and Jungbacker and Koopman (2005). In this method,
the returns are modelled without the log-squared transformation. Durbin and Koopman
(1997) showed that the loglikelihood of the state space models with non-Gaussian errors
can be written as a sum of the loglikelihood of the approximating Gaussian model and
a correction for the departures from the Gaussian assumptions with respect to the true
model. This form of likelihood has the advantage that the simulations are only required
for the departures of the likelihood of the true model from the Gaussian likelihood,
rather than for the likelihood itself. Jungbacker and Koopman (2006) used this approach
to estimate three multivariate stochastic volatility (MSV) models: the stochastic time
varying scaling factor model, where the variance matrix of the returns are scaled by
the log-volatilities, the constant correlation MSV model of Harvey et al. (1994) and a
time varying correlation MSV model based on Cholesky decomposition. In the latter set
up, the correlation dynamics is driven by the volatilities and a correlation parameter.
Tsay (2005) adopted a Cholesky decomposition based approach to ensure the positive
de�niteness of the covariance matrix. The MSV model he proposed is basically the same
time varying correlation MSV model as considered in Jungbacker and Koopman (2006)
with the correlation parameter following a stochastic autoregressive process.
Finally, the Monte Carlo Markov Chain (MCMC) methods are receiving much atten-
tion since they provide the most e¢ cient estimation tools (see Andersen et al. (1999)).
For a survey on MCMC methods and MCMC estimation of several MSV models, see
Asai et al. (2006), Meyer and Yu (2000), Chib et al. (2009). MCMC method will be
outside the scope of this paper.
When �tting an MSV model to a �nancial time series, researchers are ultimately
interested in estimating the underlying volatilities and correlations. Therefore, when
making a comparison of performances between di¤erent estimators, one should also con-
sider looking at their relative performances in estimating the in-sample volatilities and
correlations. In this respect, we employ several Monte Carlo (MC) experiments where
the performances of QML and MCL methods in estimating the parameters, volatilities
and correlations are compared. It is already known that MCL methods have better small
sample properties compared to QML methods in parameter estimation. However, in the
literature there is a need for Monte Carlo simulation studies comparing QML and MCL
methods in terms of in-sample volatility and correlation estimations in a multivariate
setup and for di¤erent parameter sets. In this paper, we attempt to �ll this gap with a
number of MC experiments for several models.
For our MC experiments, we �rst consider the Constant Correlation MSV model
of Harvey et al. (1994). As pointed out by Tsui and Yu (1999), the correlations do
not have to be constant for certain assets. This is also observed in Figure 2.1 that the
79
estimated correlations are changing over time. For this reason, we also consider the
Time Varying Correlation MSV model discussed in Jungbacker and Koopman (2006).
Another stylized fact is the so called leverage e¤ect which refers to the negative re-
lation between the current returns and future volatilities; i.e. negative returns imply
an increased leverage of the �rms which is believed to increase uncertainty and hence
volatility. As an example, in Figure 2.1, we see that on average the volatilities between
t = 800 and t = 1100, where the indices are in general falling, are much higher than the
volatilities of the period between t = 1100 and t = 1600, where the indices are in general
rising. Jungbacker and Koopman (2005) proposed a univariate SV model with leverage
and discussed how to estimate it via MCL method. In our paper we propose a direct
multivariate generalization of this model and refer to it as MSV with diagonal leverage,
where the correlations between the innovations of returns and volatilities are diagonal. A
similar but more restrictive model has been proposed, but not estimated, by Danielsson
(1998), where these correlations are modelled as a function of the variances of the inno-
vations in the volatility equations. Asai and McAleer (2006) estimated the MSV with
leverage model of Danielsson (1998) via MCL method of Sandman and Koopman (1998)
and they provided the log-squared transformation of the model necessary to implement
this estimation. Using the transformations they provided, it is also possible to esti-
mate MSV with leverage model of Danielsson (1998) with QML method. Furthermore,
we propose the MSV with non-diagonal leverage model where the correlations between
the innovations of returns and volatilities are non-diagonal; i.e. the innovations of the
volatility of series i is correlated with the innovations of the returns of series j. We also
provide the necessary transformations to estimate these two MSV with leverage models
via MCL method which are derived based on the univariate estimation in Jungbacker
and Koopman (2006). We adapt the transformations of Asai and McAleer (2006) for
estimating our two MSV models with leverages via QML method.
The results obtained in this paper con�rm that QML estimator has lower small
sample performance than MCL estimator. When the correlations are constant, the
QML estimator is performing closer to the MCL estimator especially when the true
value of the underlying correlation is high and/or if the variances of the SV processes are
high. Also, when the correlations are let to vary over time, the performance of the QML
estimator approaches to that of the MCL estimator even with lower correlations. On
the other hand, with low constant correlations and low variances of the SV processes,
the e¢ ciency of the QML estimator is relatively lower. When leverage is allowed in
the model, the performance of QML estimator is worse in estimating the underlying
correlations compared to its performance in the model without leverage. Higher values in
the true leverage matrix decreased the performance of QML estimator of the correlations
80
even more. From our results, we conclude that the QML estimator could be used when
the series are expected to have high correlations (whether constant or time varying)
and when the variances of the SV processes are high. Particularly in the case of MSV
models with leverage we do not recommend the use of QML estimator. On the other
hand, it could be of interest to estimate models with high number of series. In these
cases, the implementation of QML estimator is easier and more feasible than that of
MCL estimator. Moreover, the analytical derivatives needed for the MCL estimation are
harder to obtain with large cross-sections. One could choose to use numerical derivatives,
but the derivatives obtained by numerical approximation for large state vectors could
be very time consuming and numerically unstable. Therefore we come to the conclusion
that when estimating MSV models for several series, such as modelling the returns
of international stock markets, MCL method should be preferred for all the models
considered in this paper. The QML method could be used for the estimation of models
with medium-to-large number of series, such as the returns of a high number of assets
in a stock market, especially when the series are expected to be highly correlated with
high variances in the SV processes.
The paper is organized as follows: in Section 2.2 we discuss brie�y the Constant
Correlation MSVmodel, Time Varying Correlation MSVmodel and the two MSVmodels
with leverage we propose and later provide information on how these models can be
estimated via Quasi Maximum Likelihood and Monte Carlo Likelihood methods. In
Section 2.3 we explain the set up of our Monte Carlo experiments and discuss the results.
In section 2.4, we estimate a trivariate MSV model with leverage for the returns on
three major European stock markets. Finally in section 2.5, we discuss further topics
for research and conclude.
2.2 Multivariate Stochastic Volatility (MSV) Mod-
els
2.2.1 The Basic Model
The univariate SV model was proposed by, among others, Taylor (1982, 1986). Harvey
et al. (1994) extended this univariate SV model to a multivariate context, proposing
the �rst multivariate SV (MSV) model. If we let yt = (y1t; y2t; :::; ykt)0 be a kx1 vector
of observations at time t and ht = (h1t; h2t; :::; hkt)0 be the corresponding log-volatilities,
then this model is de�ned as:
81
yt = H1=2t "t (2.1)
Ht = diag fexp(h1t); exp(h2t); :::; exp(hkt)g = diag fexp(ht)g
ht+1 = � + �ht + �t (2.2)
h1 v N ((Ik � �)�1�;�0)
"t
�t
!v N
0;
"P" 0
0 Q�
#!(2.3)
where � is a kx1 vector of, and � is a kxk matrix of parameters. Ik denotes a kxk
identity matrix. The covariance matrices P" and Q� are of the corresponding errors "tand �t. The diagonal elements of P" are restricted to be equal to one for identi�cation
purposes, therefore P" is a correlation matrix. For simplicity, we do not consider volatil-
ity spillovers, i.e. � is a diagonal matrix. However, the volatilities ht are still dependent
on each other via Q� matrix. Finally, the (i; j) element of �0 is the (i; j) element of
Q� divided by (1��ii�jj).1 By construction, this model assumes constant correlations,therefore following Yu and Meyer (2006), we will refer to this model as Constant Cor-
relation MSV (CCMSV) model. In our analysis, we focus on the parameters, in order:
= (vecl(P")0;�0; diag(�)0; vech(Q�)
0)0.2 In this model there are k2 + 2k parameters to
estimate.
2.2.2 Time Varying Correlation MSV
The Time Varying Correlation MSV model considered in our paper is the one mentioned
in Jungbacker and Koopman (2006). We will refer to this model as TVCMSV. Following
the notation above, the observation equation (2.1) is modi�ed as:
yt = DH1=2t "t (2.4)
"t v N(0; Ik)
1That is, �0 satis�es the stationarity condition: �0 = ��0�+Q�. Therefore the elements of �0 can
be obtained by: vec(�0) = (Ik2 � � �)�1vec(Q�); where vec is the operator that stacks the columnsof a matrix and is a Kronecker product.
2The operator vec stacks all columns of a matrix, while vech stacks the columns of the lower triangular
part of a matrix and vecl stacks the columns of the strict lower triangular (exluding the leading diagonal
from the lower triangular matrix) part of a matrix.
82
where D is a lower unity triangular matrix. The idea is to decompose the conditional
variance of yt, V ar(ytjht) = Vt = DHtD0, and therefore having a stochastic dynamics
behind the variances and correlations implied by Vt. If we would call gii;t = exp(hi;t) and
D = fqij 6= 0 when i > j; 0 otherwiseg, then the implied correlations by the model aregiven by:
�ii;t =
iXs=1
q2isgss;t; i = 1; 2; :::; k
�ij;t =
jXs=1
qisqjsgss;t; i > j; i = 2; 3; :::; k
pij;t =�ij;tp�ii;t�jj;t
=
jXs=1
qisqjsgss;tvuut iXs=1
q2isgss;t
jXs=1
q2jsgss;t
This model is also a special case of factor MSV models proposed by Shephard (1996)
and further studied in Aguilar and West (2000) and Chib et.al. (2006) with the number
of factors being equal to the number of series. A shortcoming of this model is that the
driving forces underlying the volatility and correlation dynamics are the same; gii;t and
qij. The model parameters are = (vecl(D)0;�0; diag(�)0; vech(Q�)0)0. The number of
parameters to be estimated in this model is also given by k2 + 2k.
Tsay (2005) let the correlation parameters to be dynamic in the sense that the unity
lower triangular matrix D becomes Dt = fqijt 6= 0 when i > j; 0 otherwiseg whereqijt follows a Gaussian AR(1) process. Then the equation (2.4) becomes:
yt = DtH1=2t "t; (2.5)
where the kx1 vector qt evolves with the equation:
qt+1 = � +qt + vt
q1 v N ((Ik �)�1�;�0)
such that:
83
0B@ "t
�tvt
1CA v N
0B@0;264 Ik 0 0
0 Q� 0
0 0 �v
3751CA
where �0 is de�ned similar to �0. We can put this model to a state space form as
follows: let �t = (h0t; qt0)0; !t = ((�t)
0; (vt)0)0; �� = (�0; �0)0 such that:
�t+1 = �� +
� 0
0
!�t + !t;where !t v N
0;
"Q� 0
0 �v
#!(2.6)
�1 v N
(Ik � �)�1�(Ik �)�1�
!;
"�0 0
0 �0
#!
How to estimate the TVCMSV model de�ned via (2.5) and (2.6) via QML and MCL
method is left for future research. The model parameters are = (�0; diag()0;�0; diag(�)0;
vech(Q�)0)0 and the number of parameters to estimate in this model is k2 + 5k: In our
MC experiments we only consider the TVCMSV model of Jungbacker and Koopman
(2006).
2.2.3 MSV with Leverage E¤ect
The �rst MSV model with diagonal leverage we propose here is a direct generalization
of the univariate model considered in Jungbacker and Koopman (2005). Changing the
de�nition of the errors slightly, we could rewrite the equations (2.1), (2.2) and (2.3) of
CCMSV model as follows:
yt = H1=2t P �" "t (2.7)
ht+1 = � + �ht +Q���t
with the following modi�cation is made the CCMSV model:
"t
�t
!v N
0;
"Ik L
L Ik
#!(2.8)
84
where L = f�ii; i = 1:::k : �ii � [�1; 1]g is assumed to be a diagonal matrix. Thereforeby construction, the MSV with diagonal leverage model de�ned by equations (2.7) and
(2.8) implies constant correlations. A transformation similar to the one in Jungbacker
and Koopman (2005) could be then adapted to write this model in a state space form:
yt = H1=2t P �" f "�t + S�2tg (2.9)
ht+1 = � + �ht +Q�� f�1t + �2tg0B@ "�t�1t�2t
1CA v N
0B@0;264 Ik � jLj 0 0
0 Ik � jLj 0
0 0 jLj
3751CA
where S matrix is a diagonal matrix of the signs of each element of L while jLj isthe absolute value of (the elements of) L matrix. (Therefore SjLj = L). P �" and Q
��
are obtained via Cholesky defactorization of P" and Q�, respectively. The errors are
all mutually and serially independent. It can be shown that the transformed model in
equation (2.9) is consistent with the MSV model with leverage de�ned by equations (2.7)
and (2.8).
De�ning the state and signal vectors as �t = (h0t; (Q���2;t)
0)0, �t = ((Q���1;t)
0; (Q���2;t+1)0)0
and �� = (�0; 0k)0, we have the transformed model ready for MCL estimation:
yt = H1=2t P �" f "�t + S�2tg (2.10)
�t+1 = �� +
� Ik
0 0
!�t + �t; where �t v N
0;
"Q��(Ik � jLj)Q�0� 0
0 Q��jLjQ�0�
#!(2.11)
�1 v N
(Ik � �)�1��0
!;0
!
vec(0) =
"I4k2 �
� Ik
0 0
! � Ik
0 0
!#�1vec
"Q��(Ik � jLj)Q�0� 0
0 Q��jLjQ�0�
#
The parameter vector to be estimated is therefore = (vecl(P");�0; diag(�)0; vech(Q�)0;
diag(L)0)0 and the number of parameters to estimate in this model is k2 + 3k. A similar
but more restricted model is considered in Danielsson (1998) and estimated in Asai and
McAleer (2006) where �L = diag(�1�1=211 ; �2�
1=222 ; :::; �k�
1=2kk ) and Q� = f��;ijg. It should
be noted that in relation to our model, �L = Q��LP�" :
85
It could also be the case that the L matrix is non-diagonal in the sense that the errors
in the observation equation of series i are correlated with the errors in the volatility
equation of series j. Then the transformation above should be modi�ed. Assuming that
L matrix is symmetric and (positive or negative) semi-de�nite, we can de�ne a scalar s
which takes a value 1 (�1) if the L matrix is positive (negative) semi-de�nite. Thereforereplacing the S matrix with the scalar s and jLj with sL in the equations above wouldprovide us with the necessary transformation.
yt = H1=2t P �" f "�t + s�2tg
�t+1 = �� +
� Ik
0k 0k
!�t + �t; where �t v N
0;
"Q��(Ik � sL)Q�0� 0
0 Q��(sL)Q�0�
#!
�1 v N
(Ik � �)�1��0
!;0
!
vec(0)=
"I4k2 �
� Ik
0k 0k
! � Ik
0k 0k
!#�1vec
"Q��(Ik � sL)Q�0� 0
0 Q��(sL)Q�0�
#
where �� is de�ned as above. The parameter vector in this case is = (vecl(P");�0;
diag(�)0; vech(Q�)0; vec(L)0)0 which has k2 + 3k + k(k � 1)=2 parameters to estimate.
The estimation of these MSV with leverage models via QML could be done by adopt-
ing the transformations in Asai and McAleer (2006) and is discussed in section 2:4:1.
We assume throughout the paper for simplicity that whenever the true L matrix is non-
diagonal, it is symmetric. In reality, this is not necessarily the case. Moreover, the
symmetricity assumption is not needed for QML estimation but is required for MCL
estimation along with the assumption that L is positive or negative semi-de�nite.
2.2.4 Estimating the MSV Models
The estimation methods considered in this paper are the Quasi-maximum Likelihood
(QML) method of Harvey et al. (1994) and Monte Carlo Likelihood (MCL) method
of Jungbacker and Koopman (2006). These estimation methods are brie�y explained
below. Originally the CCMSV model proposed in Harvey et al. (1994) was estimated by
Quasi-maximum Likelihood approach while Jungbacker and Koopman (2006) estimated
this model by Monte Carlo Likelihood approach. The TVCMSV model with determin-
istic correlation parameter in Jungbacker and Koopman (2006) was estimated via MCL
approach. The univariate MSV model with leverage in Jungbacker and Koopman (2005)
86
was estimated using MCL method while Asai and McAleer (2006) estimated the re-
stricted model of Danielsson (1998) by the MCL approach of Sandman and Koopman
(1998). In this paper, we estimate all the models mentioned by both QML approach of
Harvey et al. (1994) and MCL approach of Jungbacker and Koopman (2006).
2.2.4.1 Quasi-maximum Likelihood (QML) Estimation
In this estimation method, the multivariate return vector yt is put through a log-squared
transformation in order to obtain a state space formation (SSF) of the model. For the
CCMSV model; the observation equation and the state equation are given as:
log(y2t ) = ht + log("2t ) = �1:2703�+ ht + �t
ht+1 = � + �ht + �t
where � is a vector of ones and the mean of log("2it) is known to be �1:2703, and itsvariance is �2=2. In fact, the distribution of log("2it) is based on a ln�
21 distribution. (See
for Sandman and Koopman (1998) for the univariate and Asai and McAleer (2006) for the
multivariate model). We can replace log("2t )+ 1:2703� with �t whose mean is therefore a
vector of zeros and covariance matrix is given by P�; which is de�ned below. QMLmethod
approximates the distribution of �t with N(0; P�). The estimation procedure is relatively
easy: Kalman �lter is applied to the log-squared returns and afterwards, the one-step
ahead prediction errors and their variances are used to obtain the likelihood function.
However, this estimation only yields minimum mean square linear estimators because
Kalman �lter is a linear �lter. How to improve the performance of QML estimators in
a multivariate setting using a nonlinear �lter is an interesting topic for future research.3
Taking into account the non-Gaussian distribution of �t, the asymptotic standard errors
can be obtained following Dunsmuir (1979). Harvey (1989, pp 212-3) notes that these
asymptotic standard errors can not be used for testing if the parameters in the matrix Q�are signi�cantly di¤erent from zero. On the other hand usual quasi-maximum likelihood
theory applies and the Bollerslev-Wooldridge robust standard errors can be used. To
estimate the in-sample estimates of volatilities and correlations, a Kalman smoothing
algorithm is employed.
Although, the QML method provides consistent estimators, because of the Gaussian
approximation, it is likely to have poor small sample properties. Breidt and Carriquiry
3Watanabe (1999) used a nonlinear �ltering to improve the performance of QML estimators in a
univariate setting.
87
(1996) and Sandman and Koopman (1998) are some of the papers that document the
ine¢ ciency of QML estimation.
It was shown in Harvey et al. (1994) that the ij-th element of the covariance matrix
P� is given by (�2=2)p�ij, where p�ii = 1 and:
p�ij =2
�2
1Xs=1
(s� 1)!(1=2)ss
p2sij
where (x)s = x(x+1):::(x+s�1). After obtaining jpijj, the sign of it can be recoveredfrom the sign of the product of corresponding pair of observations, i.e. yiyj. If more
than half of the multiplications yiyj is positive, then the sign of pij is positive.
One problem with the QML estimation is the existence of inliers, i.e. due to missing
data or simply by chance some returns will be zero or very close to zero. Therefore
a log-squared transformation of this return will explode. To take care of this, several
methods are used in the literature. Kim et al. (1996) considered a transformation such
as log(y2t +c) where c = 0:001, while Fuller (1996) assumed a data driven transformation.
We follow here the transformation discussed in Sandman and Koopman (1998) where
the values of log(y2t ) which are less than �20 is set equal to �20.Ruiz (1994) and Harvey et al. (1994) suggest that the intercept of the SV process
could be obtained directly from the observations via a moment estimator, and the log-
likelihood is optimized for the rest of the parameters. This could prove useful when the
cross section is large. In fact, this approach could also be used for the MCL estimation
when the errors are assumed to be Gaussian as in QML estimation. However, in this
paper we preferred to estimate all parameters by maximizing the loglikelihood.
The estimation of TVCMSV model via QML method is very similar the estimation of
CCMSVmodel. It is only required that in the estimation, the log-squared transformation
should be applied toD�1yt and the resulting loglikelihood function contains an additional
term: �0:5T log(det(D)): Given that in TVCMSV set up in our paper the D matrix is
lower unity triangular, its determinant is one and therefore this additional term is equal to
zero. Alternatively the D matrix could have been de�ned as a lower triangular matrix,
with nonzero values in the leading diagonal and the intercept term in the volatility
equation, � is a vector of zeros. Then the additional term in the loglikelihood would be
di¤erent than zero. See Jungbacker and Koopman (2006) for details.
For estimating the MSV model with (diagonal or nondiagonal) leverage via QML
method, the log-squared transformation as discussed in Asai and McAleer (2006) can be
applied to the model:
88
log(y2t ) = ht + log("2t ) = ht + �t
ht+1 = � + ��t + �ht + ��t ; �
�t v N
�0;���;t
�E(��t �
0t) = �L�t
���;t = ��;t � �LP�1" �L+ �LP�1"��Pj"j � 2
���0� (sts0t)
�P�1" �L
��t =q
2��LP�1" st
�L�t =�LP�1"
hnRj"j � c
q2�
o� (st�0)
iwhere st is a vector constructed from the signs of the returns in yt vector, c =
�1:2703; and the expressions for Pj"j and Rj"j can be found in the appendix of Asai
and McAleer (2006). It should be noted once again that �L = Q��LP�0" in relation to the
construction of our leverage model. As expected, when the parameter values in L matrix
are equal to zero, the state space form representation in CCMSV is obtained. Using this
transformation, it is straightforward to estimate the MSV models with leverage by QML
method by using a properly constructed Kalman �ltering.
2.2.4.2 Monte Carlo Likelihood (MCL) Estimation
Proposed by Durbin and Koopman (1997) and Shephard and Pitt (1997), this estimation
method is based on constructing the likelihood function for general state space models
using Monte Carlo techniques. Sandman and Koopman (1998) put the log-squared
transformed returns to a linear non-Gaussian state space form and proceeds with the es-
timation taking into account the true distribution of the log-squared transformed errors.
What we refer to as the MCL method in this paper is the one proposed by Jungbacker
and Koopman (2006), which extended the method in Durbin and Koopman (1997) for
the observation vector without the log-squared transformation. Other simulated max-
imum likelihood methods are considered by Danielsson and Richard (1993), Liesenfeld
and Richard (2003). In MCL method, the loglikelihood function is approximated as a
sum of a Gaussian part, constructed via Kalman �lter, and a minor remainder part which
is evaluated using simulations. Therefore it only needs a small number of simulations to
achieve the desirable accuracy for empirical analysis.
After some manipulations Durbin and Koopman (1997) showed that the likelihood
function for the non-Gaussian model based on importance sampling can be written by:
p(y) = pG(y)
Zp(yjh)p(h)pG(yjh)pG(h)
pG(hjy)dh
89
where pG(y) represents the Gaussian likelihood function of the approximating model
which is de�ned by:
~yt = ht + vt where vt s N(0; Gt) for t = 1:::N
and ht is de�ned as before. If we would de�ne _p(ytjht) = @ log p(ytjht)@ht
and �p(ytjht) =@2 log p(ytjht)
@ht@h0t, then Gt = ��p(ytjht)�1 for t=1...N and h is the mode of p(hjy). In this paper,
the models considered have p(h) = pG(h), therefore further simpli�cation can be done
on the likelihood.
By de�ning ~yt = ht+Gt _p(ytjht), it can be shown that the �rst and second derivativesof log p(hjy) and log pG(hj~y) agree in the mode h. Using the algorithm in Jungbacker
and Koopman (2006) based on Kalman �ltering and smoothing, one can compute this
mode. (See Jungbacker and Koopman (2006) for an illustration with a univariate SV
model) Later the Monte Carlo estimator of the likelihood is then given by:
p(y) = pG(~y)M�1
MXm=1
w(�m) where w(�m) =p(yjh)pG(~yjh)
and �m s pG(hj~y)
whereM is the number of samples to be generated from pG(hj~y) using the simulationsmoother algorithm of Jong and Shephard (1995) or Durbin and Koopman (2002). How-
ever, it was noted in Jungbacker and Koopman (2005) that, when Gt = ��p(ytjht)�1 isnot positive de�nite, the simulation smoothing method of Durbin and Koopman (2002)
cannot be used. In our estimations we take the number of draws M = 200.
In the case of CCMSV model, �rst and second derivatives _p(ytjht) and �p(ytjht) canbe obtained from the conditional density:
log p(ytjht) = �0:5k log(2�)� 0:5kXi=1
hit � 0:5 log(det(P"))� 0:5d0tP�1" dt for t = 1:::T
where dt = H�1=2t yt. The possible existence of an inde�nite matrix for �p(ytjht) re-
quires the approach of Jungbacker and Koopman (2005). As Jungbacker and Koopman
(2006) suggested, when the model gets too complicated or when explicit expressions for
_p(ytjht) and �p(ytjht) can not be obtained analytically, as a last resort numerical approx-imations can be used. For the CCMSV model the analytical derivatives are provided by
Jungbacker and Koopman (2006) and these can also be used to obtain the derivatives
for TVCMSV. In our estimations, we used analytical derivatives also for the MSV with
leverage models and we provide them in the appendix.
90
Finally, the in-sample estimates of the underlying volatilities can be obtained from
the smoothed estimate of the state vector � (which is just volatilities in case of CCMSV
and TVCMSV models but a larger vector in MSV models with leverage) which can be
computed from:
� =
PMi=1 �
iw(�i)PMi=1 �
i
where �i is a draw from the conditional density pG(�jy) for the approximatingGaussian linear model. When making these draws, the simulation device mentioned
in Jungbacker and Koopman (2005) is used to increase the computational e¢ ciency.
This device is based on an unconditional draw from p(�) and on a conditional mean
adjustment. (See Jungbacker and Koopman (2005) for details.)
In our experience, the computational time required for MCL estimation turned out
to be very high compared to that of the QML estimation. Especially when the sample
size or the cross-section size is increased, it takes our code much more time to converge
than it does for QML estimation. Also, when the cross-section size is large, it is not
that obvious to write the analytical derivatives and if instead one considers numerical
derivatives in this case, then the derivatives calculated with respect to large state vectors
could be very time consuming and numerically unstable. The QML method on the other
hand is much more �exible. High cross-section size or a large sample wouldn�t be a
problem for QML estimation as it would for MCL estimation. Therefore, based on our
experience, MCL estimation would be a good method for estimating MSV models for
small number of series, like returns of international stock market indices, while QML
estimation could be used for the returns of medium-to-large number of assets in a stock
market.
2.3 Monte Carlo Experiments
In this section, we report the results of our MC experiments in order to compare the
performance of QML and MCL methods when estimating the models considered in the
paper for several di¤erent parameter sets. For each model and parameter set, we gen-
erated B = 100 time series vectors of dimension k = 2 with sample size T = 500. For
comparison purposes, we look at the performances in parameter estimation as well as in
in-sample smoothed volatility and correlation estimations. The results are reported in
terms of MC means of parameter estimates, corresponding MC standard deviations and
91
root mean squared error (RMSE) for each parameter estimate as a measure of e¢ ciency.4
On the other hand, the kernel density estimates of the average deviations of estimated
volatilities, bhit, and correlations, bpt, from their true values are provided. These average
deviations are calculated for each series over the number of simulations B; i:e: for each
t:
�chit = 1B
PBb=1
�bhit;b � hit;b
��bpt = 1
B
PBb=1 (bpt;b � pt;b)
Given that in the case of Constant Correlation MSV (CCMSV) and MSVmodels with
leverage the correlations are constant (the correlation estimate is actually a parameter
estimate), the kernel density estimate of the deviations of B di¤erent estimates of the
correlation parameter from the true correlation parameter will be plotted. However, for
the Time Varying Correlation MSV (TVCMSV), as in the case of volatilities, the kernel
density estimate of �bpt is plotted.For the CCMSVmodel, the true values of the parameters = (vecl(P")0;�0; diag(�)0;
vech(Q�)0)0 and parameter estimation results are given in Table 2.1 and 2.2. These results
for the CCMSVmodel con�rm the previous results in the literature that the small sample
performance of MCL is better than that of QML; the QML method is less e¢ cient. The
e¢ ciency of QML estimator of the correlation parameter increases as the two series
become more correlated. When the series are less correlated, the QML doesn�t estimate
the correlation parameter very accurately: even though the mean is more or less around
the true value, we observe a relatively high variance. Also when the variance of the SV
processes are higher (comparing Exp 1 and Exp 3) the QML estimator gains e¢ ciency
in estimating the autoregressive coe¢ cients �: The same can be said also for the MCL
estimator of � that the RMSE is smaller when the SV processes have more variance.
Comparing Exp 1 and Exp 4, we can say that when the true value of p is high, QML and
MCL estimates of this parameter have less MC standard deviation. It is also noticed that
overall the performance of MCL estimator improves consistently for all the parameters
when p increases. When the variance of SV process is higher, it is seen that the estimation
performance of both QML and MCL estimators for autoregressive parameters increase
while there are slight changes in the RMSE of the correlation estimates. (Comparing
Exp 2 and Exp 5, Exp 6 and Exp 8)
Figure 2.2-2.3 shows the kernel density estimates of the deviations of volatility and
correlation estimates from the true values. From these �gures we could visually con�rm4For comparison purposes, in case of MSV with leverage models as well we report the results for
the parameters in P" and Q� matrices, instead of reporting the results for the Cholesky factors in their
formulation (see section 2:3).
92
that MCL estimators of the volatilities are more e¢ cient compared to QML estimators.
The high variance of the QML correlation estimates can be noticed in the third column;
especially in the experiments where the true correlation parameter value is 0:2. In fact, it
is observed that the QML correlation estimate in Exp 1 is hitting to 0 most of the time.
While higher variance of the SV process errors brings with it an increase in the variance
of the estimated volatilities for both QML and MCL estimators (comparing Exp 1 with
Exp 3 and Exp 6 with Exp 8), when the series are highly correlated both estimators seem
to perform better in estimating the underlying volatilities and correlations (comparing
Exp 1, 2, 3 with Exp 4) . In Table 2.3, the RMSE of the volatility and correlation
estimates of QML and MCL estimators are given. From this table the ine¢ ciency of
QML estimation in estimating the correlation parameter when the true value is low can
be seen clearly: when the correlation parameter value is increased from 0:2 to 0:8, the
relative RMSE of QML correlation estimates improves twofolds. Looking at this table,
it can be said that QML performs closest to MCL estimator in the experiments where
the second autoregressive parameter and the variance of the SV processes are high (Exp
5 and Exp 6). On the other hand, QML estimator of the correlation parameter performs
closer to MCL estimator when the correlation parameter is high. (Exp 6) Our conclusion
from these experiments is that MCL estimation should be preferred to QML estimation.
QML estimation could be used when the series are expected to be highly correlated, the
SV processes behind the series are strong and the sample size is large.
For the experiments with TVCMSV model, the values for the parameters (except the
correlation parameter) is chosen from the experiment 1 of CCMSV model. The corre-
lation parameter values 0:2041 and 1:3333 are chosen such that the correlation between
the volatility adjusted series are 0:2 and 0:8, respectively. The parameter estimation
results in Table 2.4 suggest that QML estimator performs better in estimating the corre-
lation parameter as well as the underlying correlations with TVCMSV model than with
CCMSV model. Also, it is observed that when the correlations are higher (in Exp 2 rel-
ative to Exp 1), the MC standard deviations and RMSE of all QML estimates are less;
while the performance of MCL estimator seems to be similar in these two experiments.
Figure 2.4 shows the kernel density estimates of the deviations of QML and MCL volatil-
ity and correlation estimates from the true values for TVCMSV model. The underlying
correlations are estimated with less variance by both QML and MCL methods when the
correlations between the series are high. Looking at Table 2.5, we can see that while the
performance of QML and MCL in estimating the underlying volatilities is more or less
the same as in corresponding CCMSV experiments (Exp 1 and Exp 4), the performance
QML estimator in estimating the underlying correlations increased relative to the MCL
correlation estimator. Therefore compared to the CCMSV model, we have less concerns
93
in estimating the TVCMSV model with QML estimator rather than MCL estimator,
while we still suggest that MCL estimator should be used.
For the MSV with diagonal leverage model, all the parameter values are taken from
experiment 1 of CCMSV model. For the additional parameters that control for the
leverage, we chose L = diagf�0:2000;�0:2500g and L = diagf�0:5500;�0:6000g. InTable 2.6, we report these true values of the parameters as well as the results of the QML
and MCL estimations when the data was generated by an MSV model with diagonal
leverage. It is observed that compared with the Exp 1 of CCMSVmodel, the performance
of QML estimator has decreased when two more parameters were included to control for
the leverage while the performance of MCL estimator seems to remain similar. When the
leverage e¤ect is higher the QML estimates of the autoregressive parameters have less
standard deviation and RMSE. It is also observed that among all the experiments done,
the performance of QML estimator in estimating the correlations of this model is lower
compared to the experiments with the models without leverage. On the other hand MCL
estimator of the leverage parameters, although having less standard deviation, seems to
be deviating from the true values relatively more compared to QML estimator. Some of
these results can be con�rmed visually from Figure 2.5 where the kernel density estimates
of the deviations of QML and MCL volatility and correlation estimates from the true
values are plotted. For instance in the third column the kernel density estimate for the
QML correlation estimates have very high variance. In practice this means that for a
given data, the QML estimate of the correlation parameter could possibly have a value
far from the true value. Finally the RMSE of the volatility and correlation estimates are
reported in Table 2.7. It is observed that for both estimators, the volatility estimates
have higher variation and RMSE compared to the Exp 1 of CCMSV model and they
increase with the strength of leverage. The correlation estimates obtained via QML
estimator have 5 to 7 times higher RMSE than the MCL estimator. Including leverage
e¤ects to the model doesn�t seem to have an e¤ect on the performance of the MCL
correlation estimator.
In two other MC experiments we consider the MSVmodel with non-diagonal leverage.
The MCL estimation of this model requires that the assumption that the leverage matrix
is symmetric and positive or negative semi-de�nite. In the �rst experiment (Exp 1) we
consider a leverage matrix that is symmetric but inde�nite, therefore the restriction of
the MCL estimation is binding. In the second experiment (Exp 2), we consider a leverage
matrix that is symmetric negative de�nite. The true values of the parameters, except the
o¤-diagonal parameter of the leverage matrix, are taken from Exp 1 of the MSV model
with diagonal leverage. The QML estimation does not require symmetricity or positive
or negative semi-de�niteness assumptions although we assume that the leverage matrix is
94
symmetric. For comparison purposes, in the �rst experiment, we also estimate the same
data with QML method imposing the restrictions on the leverage matrix. The parameter
estimation results are given in Table 2.8. In the �rst experiment comparing unrestricted
and restricted QML estimation results, we see that restricted QML estimate of the o¤-
diagonal parameter of the leverage matrix is lower and the restricted QML estimates of
the leading diagonal of the leverage matrix are higher compared to the corresponding
unrestricted QML estimates. This result con�rms that the restriction was binding. Both
unrestricted and restricted QML estimates of the correlation parameter are far from the
true value. While having uniformly less bias than the unrestricted QML estimates, MCL
estimator does its best to capture the o¤-diagonal element of the leverage matrix while
the MCL estimates for the leading diagonal of the leverage matrix have more or less the
same value as the corresponding unrestricted QML ones.
It is observed that MCL correlation estimator has similar performance in this experi-
ment compared to Exp 1 of the CCMSV model while the unrestricted or restricted QML
estimates of the correlation parameter are far from the true value with higher RMSE
compared to Exp 1 of the CCMSV model. In the second experiment (Exp 2) only the
unrestricted QML estimation results are reported along with the MCL estimation re-
sults. When the o¤-diagonal element of the leverage matrix was decreased (from Exp
1 to Exp 2), in general less bias and RMSE were obtained for the MCL estimates. As
it was in the �rst experiment, the QML estimate of the correlation parameter has very
high standard deviation. Figure 2.6 reports the kernel density estimates of the devia-
tions of unrestricted QML, restricted QML (for Exp. 1 only) and MCL volatility and
correlation estimates from the true values. Whether for volatilities or correlations, the
kernel densities corresponding to unrestricted and restricted QML estimators seem to
be very close. For both series the mode of the kernel density estimate corresponding
to the MCL volatility estimates slightly deviate from zero in the �rst experiment while
this deviation is very small or non-existent in the second experiment. This could be the
result of the restriction imposed or simply due to randomness because in the �rst exper-
iment both unrestricted and restricted QML estimates of the volatility of second series
also seem to be underestimating the true volatility. In the third column we see that the
QML correlation estimates have very high variance as in Exp. 1 of CCMSV model, while
the MCL correlation estimates have much less variance and are concentrated around the
true value of the correlation parameter. When the o¤-diagonal element of the leverage
matrix has less magnitude, the MCL correlation estimates are more dense around the
true value, while the QML correlation estimates seem to have a similar distribution as in
the �rst experiment. In Table 2.9, we provide the RMSE of the volatility and correlation
estimates.
95
Compared to the Exp. 1 of the MSV model with diagonal leverage, in the �rst
experiment the RMSE of the QML volatility estimates are higher while the RMSE of the
MCL volatility estimates are slightly lower. In the second experiment, the RMSE of the
MCL volatility and correlation estimates are lower relative to the corresponding RMSE in
Exp 1 of the MSV model with diagonal leverage. Overall, the restricted QML estimator
seems to perform closer to the MCL estimator given that the same restriction is imposed.
It is also observed that the relative RMSE of the correlation estimates increased from the
�rst experiment to second experiment. Finally, it should be noted that both QML and
MCL estimators are more e¢ cient in estimating the volatilities and correlations when
the true value of the o¤-diagonal of the leverage matrix is lower, while the improvement
is larger for the MCL estimator. Also, MCL estimator of the volatilities and correlations
in the �rst experiment with non-diagonal inde�nite leverage matrix perform similar to
the �rst experiment with diagonal leverage matrix in Table 2.7. Therefore although the
restriction imposed in the MCL method could cause underestimation of the o¤-diagonal
element of the leverage matrix as seen in Table 2.8, the volatilities and correlations are
estimated by MCL with less RMSE compared to the corresponding QML estimator.
Looking at the results of the experiments with MSV models with (diagonal and non-
diagonal) leverage and considering the high RMSE of the QML correlation estimates,
we suggest using the MCL method for estimating these models rather than using QML
estimation. It could be that the performance of QML estimator improves with higher
correlation in the data or stronger SV processes but it is not expected to be better than
the cases considered in the experiments with CCMSV model.
2.4 An Empirical Application
In this section our aim is to �nd empirical evidence supporting the MSV with non-
diagonal leverage model, i.e. the return shocks of one series is correlated with the
volatility shocks of another series. For this estimation, a trivariate series of length 1717 is
obtained from the returns of IBEX 35, FTSE 100 and DAX stock markets for the period
between 4/1/2005 and 4/11/2011. The returns are calculated as: 100 x log(Pt=Pt�1). The
descriptive statistics of the data is provided in Table 2.10. It is observed that the IBEX
35 and DAX returns are skewed right while FTSE-100 is skewed left. On the other hand,
as expected, all series have high kurtosis. We also report the Box-Ljung statistics for the
returns and its squared and log-squared transformations. The 5% critical value with ten
degrees of freedom in a chi-square distribution corresponds to 18.3. Box-Ljung statistic
for the return series, yt, suggests that the data may not be random walks, more likely
in the case of FTSE-100. On the other hand, there is strong evidence of nonlinearity in
96
the squared returns and log-squared returns; suggesting that there is autocorrelation in
these series.
A univariate SV model with leverage is �t for each of the series. The QML and MCL
estimation results for the univariate model is given for each series in Table 2.11. From
the results of the univariate estimation, we see that the MCL estimates imply more
persistent SV processes compared to QML estimates. MCL estimates of the autoregres-
sive parameter suggest that these SV processes are even close to random walk. This is
also con�rmed by the Box-Ljung statistic for the return series, in Table 2.10 Also it is
noted that MCL estimates of the leverage coe¢ cients, L; are higher compared to QML
estimates.
The estimation of the MSV model with leverage requires the restriction that the
L matrix is symmetric and (positive or negative) semide�nite. This latter restriction
is not required by the QML estimation. However for comparison purposes, we also
estimated the model via QML assuming this restriction. The estimation results for the
MSV model with leverage are given in Table 2.12. If we compare the results of the
multivariate estimation with the results of the univariate estimation in Table 2.11, we
see that the QML estimates of the intercept, of the autoregressive parameter and of the
variance of the SV process are more or less the same in both cases while the self-leverage
of each series, that is the diagonal of L matrix, is estimated to be less in magnitude for
FTSE-100 and DAX indices compared to the univariate results.
The MCL estimates of the autoregressive parameters are higher in the univariate es-
timation compared to the multivariate estimation while the estimates of the self-leverage
of each series are lower in the multivariate estimation. When comparing the unrestricted
QML and MCL estimation results, we see that the correlation estimates obtained by
these two methods are more or less the same. While the MCL estimates of the autore-
gressive parameters are higher, the MCL estimates of elements of the variance matrix of
the SV process are lower; this is due to the fact that the estimation tries to match the
unconditional variance in the data and when the estimates of the autoregressive parame-
ters are high, the variance matrix of the SV process is pushed downwards. When we look
at the leverage matrix estimates, we see that MCL estimates of the diagonal elements
of L matrix are higher compared to the QML estimates. The MCL leverage parameter
estimates are statistically signi�cant. Moreover, the likelihood ratio test to compare the
MCL estimation results of CCMSV and MSV-NDL models suggest that the data is ex-
plained better by the latter model.5 Figure 2.7 shows for each series the absolute values
of the returns plotted along with the QML and MCL smooth estimates of the standard
5The likelihood ratio test can�t be used with the QML estimation because it is based on approxima-
tions.
97
deviations. It is observed that QML overestimated the volatilities of the period after
the big volatility shocks. (for example around t = 1000) Finally, the MCL estimates
of the standard deviations follow the absolute values of the returns closely while QML
estimates are experiencing some jumps when volatility of the data is increasing.
2.5 Conclusions
In this paper, via Monte Carlo (MC) experiments, we compare the performance of Quasi
Maximum Likelihood estimation method of Harvey et al. (1994) and Monte Carlo Like-
lihood estimation method of Jungbacker and Koopman (2006) in estimating the para-
meters as well as in estimating the underlying volatilities and correlations. With these
methods, we estimate the Constant Correlation MSV model of Harvey et al. (1994), the
Time Varying Correlation MSV model of Jungbacker and Koopman (2006). Moreover,
we propose two MSVmodels with leverage which are new to the literature. The �rst MSV
model with leverage is a direct generalization of the univariate model in Jungbacker and
Koopman (2005). In this model, each series has its own leverage e¤ect: i.e. the return
shocks of series i is correlated with the volatility shocks of series i while the correlation of
the return shocks of series i and the volatility shocks of series j is zero. Therefore in this
model the leverage matrix is diagonal, hence we refer to it as MSV model with diagonal
leverage. In the second MSV model with leverage, we relax this assumption and let the
o¤-diagonals of the leverage matrix to be non-zero. We refer this model as MSV with
non-diagonal leverage.
The estimation of CCMSV model via QML and MCL are discussed in Harvey et al.
(1994) and Jungbacker and Koopman (2006), respectively. Jungbacker and Koopman
(2006) also provides the estimation procedure for the TVCMSVmodel which follows from
a small modi�cation of the estimation procedure of CCMSV model. This modi�cation
can be applied to the QML method in order to estimate the TVCMSV model. For
the estimation of MSV models with (diagonal and non-diagonal) leverage, we adopt
the transformations discussed in Asai and McAleer (2006). On the other hand, the
transformations of the MSV models with leverage given in this paper are new to the
literature and are based on the univariate transformation in Jungbacker and Koopman
(2006).
We considered eight di¤erent parameter sets for the CCMSV model in our MC exper-
iments. The results con�rm the previous �ndings in the literature that QML estimator
is ine¢ cient in terms of parameter estimation. It is observed that when the true value
of the correlation parameter is low, the QML estimator of this parameter has very high
variance. Therefore, when estimating a model with real data, if the underlying correla-
98
tion parameter is low, the QML estimate will not be very informative. We also observed
that the performance of the QML estimator increases as the series become more cor-
related and when the SV processes have higher variance. In estimating the underlying
volatilities and correlations, the performance of MCL estimator was superior to that
of QML estimator in all parameter sets, although the QML estimator was performing
closely in the experiments where the correlations were higher or SV processes had higher
variance.
For the TVCMSV model, we considered two experiments; one with low correlation
and another one with high correlation. It appeared that the performance of the QML
estimator relative to the MCL estimator was much better compared to the experiments
of CCMSV model. With time varying correlations, the QML estimator was able to
perform close to the MCL estimator even when the correlations were low.
For the MSV models with diagonal leverage we considered two experiments, one with
low leverage and another one with high leverage. Our results showed that relative to
the experiments with CCMSV, the ine¢ ciency of the QML estimator increased while
the performance of MCL stayed the same when leverage is introduced. Increasing the
true values of the leverage parameters further decreases the performance of the QML
estimator. The correlation estimates of the QML model had very high root mean squared
error (�ve to seven times the ones of MCL estimator). For the MSV model with non-
diagonal leverage, we also considered two experiments; one where the leverage matrix
was inde�nite and another one where it was negative de�nite. In the �rst experiment,
the restriction that "the leverage matrix should be symmetric and positive or negative
semi-de�nite" was binding, while in the latter it was not. Our results con�rm that in the
�rst experiment, even though the MCL method underestimated the o¤-diagonal leverage
parameter, it was able to capture the underlying volatilities and correlations almost as
good as in the case of CCMSV model. On the other hand in both MSV models with
leverage, QML correlation estimates had high bias and high standard deviation such
that its performance was worse than in the corresponding case of CCMSV model.
Based on our results, we conclude that even though in the case of TVCMSV, QML
estimator performs close to MCL estimator, the latter is always preferred. We do not
recommend using QML estimators for the models with leverage. Although QML method
can be implemented much easier than MCL and the estimation time is much less in QML
estimation; we suggest its use if it is expected that the series have high and/or time
varying correlation and the SV processes have higher variance. Given the results in the
literature on the ine¢ ciency of QML estimator in small samples, it would be also a plus if
the sample size is large, when using QMLmethod. On the other hand the implementation
of MCL estimation is relatively more complicated than the QML estimation. Therefore
99
MCL estimation requires much more time to converge. When the cross-section size is
large, the analytical derivatives for the MCL estimation are harder to obtain and if one
would like to use numerical derivatives in this case, then the derivatives calculated for
large state vectors could be very time consuming and numerically unstable. QML is not
as much a¤ected by the large cross-sections or large sample sizes. Therefore based on
our experience, we would suggest using MCL method in the estimation of MSV models
for several series, as for modelling the returns of international stock market indices, and
QML method could be used for the estimation with medium-to-large number of series
from a stock market.
The ine¢ ciency of QMLmethod could be improved partially by employing a nonlinear
�lter instead of Kalman �lter. The latter is a linear �lter and therefore leads to minimum
mean square linear estimators rather than minimummean squared estimators. Watanabe
(1999) provides a nonlinear �lter for QML estimation for the univariate SV model and
extending it to a multivariate setup would be an interesting topic.
Another point to consider would be to introduce a correlation between the SV process
errors and the stochastic correlation parameter errors in the Tsay (2005) model. The
intuition behind this extra parameter would be that the volatility shocks are correlated
with the correlation shocks, meaning that when the series are more volatile, they are
expected to be more correlated. As we have seen in the recent crisis, the markets tend
to move more closely when there are bad news, while their recoveries from these falls
might not be as correlated.
2.6 Appendix
Following Jungbacker and Koopman (2006) and Lutkepohl (1996), we obtained the deriv-
atives for the bivariate MSV model with diagonal leverage needed for deriving the ap-
proximating linear model. For the nondiagonal leverage model, it can be easily modi�ed.
On the other hand, these derivatives are extendable to cases with more than k = 2 series;
in the empirical estimation part these derivatives are used for k = 3 case.
yt = H1=2t P �" "t =) "t = P �
�1" H
�1=2t yt
Ht and P �" as de�ned in (1) and (3). Then using (9) we can write:
dt = P ��1" H
�1=2t yt � SQ�
�1� �2;t
� de�ned as in (10): If we let X = I2�SL where I2 is a 2x2 identity matrix and �1;tbe the volatilities part of �t; then the loglikelihood for (10) would be given by:
100
log p(ytjht) = �0:5k log(2�)� 0:5P
i �1;t;i � 0:5 log(det(P �"XP �0" ))� 0:5d0tX�1dt
Then the �rst derivatives with respect to the state vector �t would be given by:
@lt@�t
= �0:5
266641
1
0
0
37775� 0:5 @d0t@�t(X�1 +X�10) dt
@d0t@�t
=�@dt@�t
�0=
0@ n�0:5P ��1" diag(H
�1=2t yt)
o0�nSQ�
�1�
o01A
The second derivatives are obtained from:
@2lt@�t@�0t
= �0:5(@d0t@�t(X�1 +X�10) @dt
@�0t+ [d0t (X
�1 +X�10) I4]@vec
�@d0t@�t
�@�0t
)
where I4 is a 4x4 identity matrix and is a Kronecker product. The last expressionin the equation is equal to:
@vec
�@d0t@�t
�@�0t
= 0:25Z
where Z1;1 =nP �
�1" diag(H
�1=2t yt)
o1;1, Z5;1 =
nP �
�1" diag(H
�1=2t yt)
o2;1, and
Z6;2 =nP �
�1" diag(H
�1=2t yt)
o2;2while the rest of the entries are zeros.
101
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105
Figure 2.1: Indices, returns, squared returns (as a proxy for volatilities) and correlations
(estimated by a rolling window of 60 days) of FTSE-100 and DAX stock markets. Source:
Yahoo Finance.
0 200 400 600 800 1000 1200 1400 16002000
4000
6000
8000
10000
Indi
ces
FT SE100 and DAX, Period: 04/01/200504/11/2011
0 200 400 600 800 1000 1200 1400 160010
5
0
5
10
Ret
urns
0 200 400 600 800 1000 1200 1400 16000
50
100
150
Sq.
Ret
urns
0 200 400 600 800 1000 1200 1400 16000.7
0.8
0.9
1
Cor
rela
tions
FTSE100DAX
106
Figure 2.2: Kernel density estimates of the deviations of MCL and QML volatility and
correlation estimates from the true ones, for the CCMSV model. Experiments 1 to 4.
0.5 0 0.50
5
10
∆h1t
Exp
1
MCLQMLzero l ine
0.5 0 0.50
5
10
∆h2t
0.5 0 0.50
10
20
30∆p
0.5 0 0.50
5
10
Exp
2
0.5 0 0.50
5
10
0.5 0 0.50
10
20
30
0.5 0 0.50
5
10
Exp
3
0.5 0 0.50
10
20
30
0.5 0 0.50
5
10
Exp
4
0.5 0 0.50
5
10
0.5 0 0.50
10
20
30
0.5 0 0.50
5
10
107
Figure 2.3: Kernel density estimates of the deviations of MCL and QML volatility and
correlation estimates from the true ones, for the CCMSV model. Experiments 5 to 8.
0.5 0 0.50
5
10
Exp
5
∆h1t
0.5 0 0.50
5
10
∆h2t
0.5 0 0.50
10
20
30∆p
0.5 0 0.50
5
10
Exp
6
0.5 0 0.50
5
10
0.5 0 0.50
10
20
30
0.5 0 0.50
5
10
Exp
7
0.5 0 0.50
5
10
0.5 0 0.50
10
20
30
MCLQMLzero l ine
0.5 0 0.50
5
10
Exp
8
0.5 0 0.50
5
10
0.5 0 0.50
10
20
30
108
Figure 2.4: Kernel density estimates of the deviations of MCL and QML volatility and
correlation estimates from the true ones, for the TVCMSV model. Experiments 1 and
2.
0.5 0 0.50
2
4
6
8
10
∆h1t
Exp
1
0.5 0 0.50
2
4
6
8
10
∆h2t
0.5 0 0.50
20
40
60
80
∆pt
MCLQMLzero l ine
0.5 0 0.50
2
4
6
8
10
Exp
2
0.5 0 0.50
2
4
6
8
10
0.5 0 0.50
20
40
60
80
109
Figure 2.5: Kernel density estimates of the deviations of MCL and QML volatility and
correlation estimates from the true ones, for the MSV model with diagonal leverage.
Experiments 1 and 2.
0.5 0 0.50
2
4
6
8
10
∆h1t
Exp
1
MCLQMLzero l ine
0.5 0 0.50
2
4
6
8
10
∆h2t
0.5 0 0.50
5
10
15
20
25
30∆p
0.5 0 0.50
2
4
6
8
10
Exp
2
0.5 0 0.50
2
4
6
8
10
0.5 0 0.50
5
10
15
20
25
30
110
Figure 2.6: Kernel density estimates of the deviations of MCL and QML volatility and
correlation estimates from the true ones, for the MSV model with non-diagonal leverage.
Experiments 1 and 2.
0.5 0 0.50
2
4
6
8
10
Exp
1
∆h1t
MCLQMLunresQMLreszero l ine
0.5 0 0.50
2
4
6
8
10
∆h2t
0.5 0 0.50
5
10
15
20
25
30∆p
0.5 0 0.50
2
4
6
8
10
Exp
2
0.5 0 0.50
2
4
6
8
10
0.5 0 0.50
5
10
15
20
25
30
111
Figure 2.7: Absolute values of the returns and the MCL and QML smooth estimates of
the standard deviations for IBEX 35, FTSE 100 and DAX stock markets
0 200 400 600 800 1000 1200 1400 16000
5
10
15
IBE
X 3
5
0 200 400 600 800 1000 1200 1400 16000
5
10
15
FTS
E 1
00
0 200 400 600 800 1000 1200 1400 16000
5
10
15
DA
X
| yt |
MCLQML
112
Table 2.1: The parameter estimation results of the simulations where the data is gen-
erated by a CC-MSV model and estimated via QML and MCL methods. For each
experiment, the true parameter values are reported in the �rst row. Then for each esti-
mation method, MC mean, standard deviation (in parantheses) and root mean squared
error (in square brackets) are reported, respectively. Experiments 1-4.Estim.nParam. fP"g21 �11 �21 �11 �22 fQ�g11 fQ�g21 fQ�g22Exp 1 - True 0.2000 -0.1000 -0.1300 0.9000 0.9500 0.1500 0.0400 0.0800
QML 0.1509 -0.1368 -0.2770 0.8646 0.8948 0.2373 0.0539 0.1824
(0.1751) (0.1092) (0.4187) (0.0971) (0.1488) (0.2731) (0.0601) (0.2454)
[0.1819] [0.1153] [0.4437] [0.1033] [0.1587] [0.2867] [0.0617] [0.2659]
MCL 0.1943 -0.1197 -0.1673 0.8811 0.9354 0.1649 0.0390 0.0862
(0.0444) (0.0617) (0.0965) (0.0499) (0.0350) (0.0695) (0.0231) (0.0363)
[0.0447] [0.0647] [0.1034] [0.0534] [0.0380] [0.0711] [0.0231] [0.0368]
Exp 2 - True 0.2000 -0.1000 -0.1300 0.9000 0.9800 0.1500 0.0400 0.0800
QML 0.1919 -0.1821 -0.2573 0.8241 0.9604 0.2593 0.0380 0.1015
(0.1858) (0.2093) (0.5369) (0.1835) (0.0798) (0.3124) (0.0532) (0.1333)
[0.1860] [0.2248] [0.5518] [0.1986] [0.0821] [0.3310] [0.0533] [0.1350]
MCL 0.1957 -0.1376 -0.1822 0.8649 0.9716 0.1797 0.0392 0.0809
(0.0427) (0.0727) (0.1018) (0.0641) (0.0157) (0.0782) (0.0288) (0.0279)
[0.0429] [0.0819] [0.1144] [0.0731] [0.0178] [0.0836] [0.0288] [0.0279]
Exp 3 - True 0.2000 -0.1000 -0.1300 0.9000 0.9500 0.4000 0.1500 0.3500
QML 0.1880 -0.1265 -0.1526 0.8707 0.9404 0.4954 0.1618 0.3649
(0.1833) (0.0790) (0.0647) (0.0577) (0.0214) (0.2785) (0.0732) (0.1333)
[0.1837] [0.0833] [0.0686] [0.0647] [0.0234] [0.2944] [0.0742] [0.1341]
MCL 0.1899 -0.1119 -0.1402 0.8867 0.9459 0.3974 0.1491 0.3325
(0.0428) (0.0536) (0.0499) (0.0346) (0.0171) (0.0881) (0.0498) (0.0850)
[0.0440] [0.0549] [0.0510] [0.0370] [0.0179] [0.0882] [0.0498] [0.0868]
Exp 4 - True 0.8000 -0.1000 -0.1300 0.9000 0.9500 0.1500 0.0400 0.0800
QML 0.8034 -0.1613 -0.2117 0.8363 0.9197 0.2747 0.0567 0.1230
(0.0365) (0.1398) (0.2018) (0.1377) (0.0660) (0.3111) (0.0568) (0.1578)
[0.0366] [0.1527] [0.2177] [0.1517] [0.0726] [0.3352] [0.0593] [0.1635]
MCL 0.7961 -0.1130 -0.1692 0.8854 0.9349 0.1559 0.0477 0.0841
(0.0160) (0.0476) (0.0880) (0.0384) (0.0333) (0.0500) (0.0227) (0.0326)
[0.0165] [0.0494] [0.0964] [0.0411] [0.0366] [0.0503] [0.0240] [0.0328]
113
Table 2.2: The parameter estimation results of the simulations where the data is gen-
erated by a CC-MSV model and estimated via QML and MCL methods. For each
experiment, the true parameter values are reported in the �rst row. Then for each esti-
mation method, MC mean, standard deviation (in parantheses) and root mean squared
error (in square brackets) are reported, respectively. Experiments 5-8.Estim.nParam. fP"g21 �11 �21 �11 �22 fQ�g11 fQ�g21 fQ�g22Exp 5 - True 0.2000 -0.1000 -0.1300 0.9000 0.9800 0.4000 0.1500 0.3500
QML 0.1876 -0.1322 -0.1873 0.8676 0.9712 0.5114 0.1554 0.3439
(0.1940) (0.0832) (0.0814) (0.0598) (0.0111) (0.2813) (0.0651) (0.0929)
[0.1944] [0.0892] [0.0995] [0.0680] [0.0142] [0.3025] [0.0653] [0.0931]
MCL 0.1918 -0.1150 -0.1727 0.8852 0.9733 0.4054 0.1464 0.3314
(0.0410) (0.0521) (0.0760) (0.0334) (0.0105) (0.0841) (0.0447) (0.0750)
[0.0418] [0.0542] [0.0872] [0.0366] [0.0124] [0.0842] [0.0449] [0.0772]
Exp 6 - True 0.8000 -0.1000 -0.1300 0.9000 0.9800 0.4000 0.1500 0.3500
QML 0.8047 -0.1304 -0.1747 0.8714 0.9725 0.4666 0.1504 0.3383
(0.0376) (0.0763) (0.0875) (0.0622) (0.0132) (0.2286) (0.0852) (0.0997)
[0.0379] [0.0821] [0.0983] [0.0684] [0.0152] [0.2381] [0.0852] [0.1004]
MCL 0.7893 -0.1107 -0.1549 0.8906 0.9756 0.3703 0.1482 0.3250
(0.0183) (0.0403) (0.0779) (0.0285) (0.0117) (0.0821) (0.0608) (0.0655)
[0.0212] [0.0417] [0.0818] [0.0300] [0.0125] [0.0873] [0.0608] [0.0701]
Exp 7 - True 0.8000 -0.1000 -0.1300 0.9000 0.9500 0.4000 0.1500 0.3500
QML 0.8097 -0.1171 -0.1726 0.8822 0.9353 0.4297 0.1439 0.3852
(0.0430) (0.0561) (0.0970) (0.0505) (0.0307) (0.2127) (0.1008) (0.1633)
[0.0441] [0.0586] [0.1059] [0.0536] [0.0340] [0.2147] [0.1010] [0.1671]
MCL 0.7901 -0.1042 -0.1412 0.8978 0.9469 0.3520 0.1422 0.3193
(0.0211) (0.0373) (0.0543) (0.0247) (0.0175) (0.0763) (0.0560) (0.0725)
[0.0233] [0.0376] [0.0555] [0.0248] [0.0178] [0.0901] [0.0565] [0.0787]
Exp 8 - True 0.8000 -0.1000 -0.1300 0.9000 0.9800 0.1500 0.0400 0.0800
QML 0.8030 -0.1294 -0.2093 0.8650 0.9676 0.2212 0.0525 0.1035
(0.0424) (0.1174) (0.1583) (0.1079) (0.0245) (0.2554) (0.0468) (0.0662)
[0.0426] [0.1210] [0.1771] [0.1134] [0.0275] [0.2652] [0.0484] [0.0702]
MCL 0.7971 -0.1097 -0.1693 0.8854 0.9737 0.1469 0.0440 0.0826
(0.0201) (0.0429) (0.0765) (0.0392) (0.0122) (0.0444) (0.0248) (0.0233)
[0.0203] [0.0440] [0.0860] [0.0419] [0.0137] [0.0445] [0.0251] [0.0235]
114
Table 2.3: Root mean squared error of the QML and MCL volatility and correlation
estimates for CC-MSV modelExperiment Method �h1t �h2t �p Experiment Method �h1t �h2t �p
Exp 1 QML 0.6409 0.5459 0.1819 Exp 5 QML 1.0889 0.8983 0.1944
MCL 0.5407 0.4951 0.0447 MCL 1.0506 0.8286 0.0418
QML/MCL 1.1853 1.1026 4.0694 QML/MCL 1.0364 1.0842 4.6507
Exp 2 QML 0.6334 0.6336 0.1860 Exp 6 QML 0.8746 0.9181 0.0379
MCL 0.5611 0.4676 0.0429 MCL 0.7893 0.7105 0.0212
QML/MCL 1.1288 1.3550 4.3357 QML/MCL 1.1081 1.2922 1.7877
Exp 3 QML 0.9383 0.8748 0.1837 Exp 7 QML 0.9336 0.9068 0.0441
MCL 0.8885 0.8137 0.0440 MCL 0.8309 0.8759 0.0233
QML/MCL 1.0560 1.0751 4.1750 QML/MCL 1.1236 1.0353 2.1724
Exp 4 QML 0.6829 0.6164 0.0366 Exp 8 QML 0.5563 0.8516 0.0426
MCL 0.5038 0.5015 0.0165 MCL 0.5217 0.6874 0.0203
QML/MCL 1.3555 1.2291 2.2182 QML/MCL 1.0663 1.2388 2.0985
Table 2.4: The parameter estimation results of the simulations where the data is gen-
erated by a TVC-MSV model and estimated via QML and MCL methods. For each
experiment, the true parameter values are reported in the �rst row. Then for each es-
timation method, MC mean, standard deviation (in paranthesis and root mean squared
error (in square brackets) are reported, respectively.Estim.nParam. fDg21 �11 �21 �11 �22 fQ�g11 fQ�g21 fQ�g22Exp 1 - True 0.2041 -0.1000 -0.1300 0.9000 0.9500 0.1500 0.0400 0.0800
QML 0.2034 -0.1852 -0.2068 0.8205 0.9178 0.2742 0.0648 0.0897
(0.0439) (0.2449) (0.3879) (0.2109) (0.1540) (0.3603) (0.0815) (0.1079)
[0.0439] [0.2593] [0.3954] [0.2254] [0.1574] [0.3811] [0.0852] [0.1083]
MCL 0.2039 -0.1322 -0.1691 0.8684 0.9352 0.1783 0.0501 0.0903
(0.0174) (0.0703) (0.0759) (0.0670) (0.0286) (0.0817) (0.0315) (0.0348)
[0.0174] [0.0773] [0.0854] [0.0741] [0.0322] [0.0865] [0.0330] [0.0363]
Exp 2 - True 1.3333 -0.1000 -0.1300 0.9000 0.9500 0.1500 0.0400 0.0800
QML 1.3275 -0.1541 -0.2205 0.8459 0.9151 0.2712 0.0597 0.0984
(0.0250) (0.1413) (0.3675) (0.1315) (0.1338) (0.3438) (0.0774) (0.0945)
[0.0257] [0.1513] [0.3785] [0.1421] [0.1383] [0.3646] [0.0798] [0.0962]
MCL 1.3359 -0.1358 -0.1677 0.8623 0.9358 0.1759 0.0441 0.0898
(0.0164) (0.0829) (0.0809) (0.0780) (0.0307) (0.0757) (0.0286) (0.0392)
[0.0166] [0.0903] [0.0892] [0.0867] [0.0338] [0.0800] [0.0289] [0.0404]
115
Table 2.5: Root mean squared error of the QML and MCL volatility and correlation
estimates TVCMSV modelExperiment n Variable Method �h1t �h2t �pt
Exp 1 QML 0.6045 0.7038 0.1439
MCL 0.5176 0.4678 0.1096
QML/MCL 1.1679 1.5044 1.3129
Exp 2 QML 0.6834 0.5564 0.0571
MCL 0.5148 0.5185 0.0475
QML/MCL 1.3275 1.0730 1.2021
Table 2.6: The parameter estimation results of the simulations where the data is gen-
erated by an MSV model with diagonal leverage and estimated via QML and MCL
methods. For each experiment, the true parameter values are reported in the �rst row.
Then for each estimation method, MC mean, standard deviation (in paranthesis) root
mean squared error (in square brackets) are reported, respectively.Estim.nParam. fP"g21 �11 �21 �11 �22 L11 L22 fQ�g11 fQ�g21 fQ�g22Exp 1 - True 0.2000 -0.1000 -0.1300 0.9000 0.9500 -0.2000 -0.2500 0.1500 0.0400 0.0800QML 0.1289 -0.1650 -0.2379 0.8431 0.9093 -0.1543 -0.2431 0.2529 0.0475 0.1306
(0.2438) (0.1745) (0.3002) (0.1472) (0.1125) (0.2192) (0.3691) (0.2728) (0.0578) (0.1547)[0.2539] [0.1862] [0.3190] [0.1579] [0.1197] [0.2240] [0.3692] [0.2916] [0.0583] [0.1628]
MCL 0.1935 -0.1272 -0.1712 0.8760 0.9350 -0.1335 -0.1822 0.1650 0.0413 0.0842(0.0499) (0.0603) (0.0864) (0.0508) (0.0322) (0.1526) (0.2087) (0.0681) (0.0277) (0.0347)[0.0503] [0.0662] [0.0957] [0.0562] [0.0355] [0.1665] [0.2195] [0.0697] [0.0277] [0.0349]
Exp 2 - True 0.2000 -0.1000 -0.1300 0.9000 0.9500 -0.5500 -0.6000 0.1500 0.0400 0.0800QML 0.0751 -0.1798 -0.2235 0.8274 0.9162 -0.3822 -0.5342 0.2520 0.0404 0.1071
(0.2748) (0.1595) (0.2049) (0.1285) (0.0700) (0.2371) (0.2939) (0.2723) (0.0504) (0.1214)[0.3018] [0.1784] [0.2252] [0.1476] [0.0777] [0.2905] [0.3012] [0.2908] [0.0504] [0.1243]
MCL 0.2029 -0.1648 -0.1976 0.8452 0.9273 -0.2952 -0.4543 0.2033 0.0446 0.0879(0.0416) (0.0696) (0.0958) (0.0616) (0.0373) (0.1380) (0.1675) (0.0733) (0.0276) (0.0329)(0.0417) [0.0951] [0.1173] [0.0824] [0.0437] [0.2898] (0.2220) (0.0906) (0.0280) (0.0338)
Table 2.7: Root mean squared error of the QML and MCL volatility and correlation
estimates of the MSV model with diagonal leverageExperiment n Variable Method �h1t �h2t �pt
Exp 1 QML 0.6821 0.6206 0.2539
MCL 0.6208 0.5414 0.0503
QML/MCL 1.0987 1.1463 5.0477
Exp 2 QML 0.7054 0.6480 0.3018
MCL 0.7231 0.5976 0.0417
QML/MCL 0.9755 1.0843 7.2374
116
Table2.8:TheparameterestimationresultsofthesimulationswherethedataisgeneratedbyanMSV
modelwithnon-diagonal
leverageandestimatedviaunrestrictedQML,restrictedQMLandMCLmethods.Experiment1referstothecasewherethe
leveragematrix,L,isinde�nitewhileinExperiment2itis(negative)de�nite.TherestrictionthatwasimposedtotheQML
estimationistheonethatisrequiredforMCLestimation,namely,theLmatrixispositiveornegativesemide�nite.Foreach
experiment,thetrueparametervaluesarereportedinthe�rstrow.Thenforeachestimationmethod,MCmean,standard
deviation(inparenthesis)androotmeansquarederror(insquarebrackets)arereported,respectively.
Estim.nParam.
fP"g 21
�11
�21
�11
�22
L11
L21
L22
fQ�g 11fQ
�g 21fQ
�g 22
Exp1-True
0.2000
-0.1000
-0.1300
0.9000
0.9500
-0.2000
-0.2300
-0.2500
0.1500
0.0400
0.0800
QML-unrestricted
0.1124
-0.1811
-0.2374
0.8292
0.9099
-0.1689
-0.1539
-0.2280
0.2844
0.0514
0.1511
(0.2428)(0.1616)(0.2617)(0.1430)(0.0984)(0.2474)(0.1795)(0.3349)(0.3023)(0.0548)(0.2548)
[0.2581]
[0.1808]
[0.2829]
[0.1596]
[0.1062]
[0.2494]
[0.1950]
[0.3357]
[0.3309]
[0.0560]
[0.2645]
QML-restricted
0.1323
-0.1964
-0.2716
0.8126
0.8970
-0.2009
-0.0951
-0.2637
0.2969
0.0672
0.1671
(0.2081)(0.1506)(0.2498)(0.1407)(0.0934)(0.2035)(0.1251)(0.2612)(0.2475)(0.0678)(0.2053)
[0.2189]
[0.1788]
[0.2871]
[0.1656]
[0.1074]
[0.2035]
[0.1330]
[0.2615]
[0.2878]
[0.0731]
[0.2231]
MCL
0.2045
-0.1550
-0.2307
0.8560
0.9141
-0.1530
-0.1236
-0.2342
0.1971
0.0447
0.1099
(0.0474)(0.0830)(0.1517)(0.0707)(0.0561)(0.1008)(0.0800)(0.1373)(0.0863)(0.0393)(0.0552)
[0.0477]
[0.0996]
[0.1821]
[0.0833]
[0.0666]
[0.1112]
[0.1087]
[0.1382]
[0.0984]
[0.0396]
[0.0628]
Exp2-True
0.2000
-0.1000
-0.1300
0.9000
0.9500
-0.2000
-0.0500
-0.2500
0.1500
0.0400
0.0800
QML-unrestricted
0.1535
-0.1560
-0.2669
0.8428
0.8957
-0.1441
-0.0106
-0.1813
0.2049
0.0426
0.1630
(0.2405)(0.1436)(0.3407)(0.1843)(0.1388)(0.2530)(0.1958)(0.3825)(0.1696)(0.0602)(0.2345)
[0.2450]
[0.1541]
[0.3672]
[0.1929]
[0.1490]
[0.2591]
[0.1997]
[0.3886]
[0.1783]
[0.0602]
[0.2488]
MCL
0.1960
-0.1356
-0.2012
0.8702
0.9232
-0.1452
-0.0159
-0.1995
0.1726
0.0445
0.1046
(0.0370)(0.0515)(0.1082)(0.0441)(0.0413)(0.1036)(0.0849)(0.1384)(0.0658)(0.0306)(0.0493)
[0.0372]
[0.0626]
[0.1296]
[0.0532]
[0.0492]
[0.1172]
[0.0915]
[0.1473]
[0.0696]
[0.0309]
[0.0551]
117
Table 2.9: Root mean squared error of the QML and MCL volatility and correlation
estimates of the MSV model with non-diagonal leverage. In Experiment 1 the leverage
matrix, L, is inde�nite while in Experiment 2 it is (negative) de�nite.Experiment n Variable Method �h1t �h2t �pt
Exp 1 QML - unres 0.7125 0.6760 0.2581
QML - res 0.7047 0.6576 0.2189
MCL 0.6164 0.5206 0.0477
QML - unres/MCL 1.1559 1.2985 5.4109
QML - res/MCL 1.1433 1.2632 4.5891
Exp 2 QML - unres 0.6787 0.6406 0.2450
MCL 0.5979 0.5098 0.0372
QML - unres/MCL 1.1351 1.2566 6.5860
Table 2.10: Descriptive statistics of the returnsStatistics n Series IBEX-35 FTSE-100 DAX
Mean -0.0034 0.0076 0.0192
SD 1.5981 1.3667 1.5163
Skewness 0.1504 -0.1385 0.0346
Kurtosis 10.7492 10.4146 9.7788
Maximum 13.4836 9.3842 10.7975
Minimum -9.5859 -9.2646 -7.4335
Box-Ljung, yt 21.49 55.67 21.97
Box-Ljung, y2t 595.21 1227.2 792.34
Box-Ljung, log y2t 484.99 398.69 318.29
Table 2.11: The empirical estimation results for the univariate SV model with leverage.Estim. Series � � L Q� Log-like AIC BIC
QML IBEX 35 0.0137 0.9636 -0.5262 0.0693 -3890.6 7789.2 7811.0
(0.0012) (0.0055) (0.0944) (0.0080)
FTSE 100 0.0003 0.9625 -0.4964 0.0747 -3911.7 7831.4 7853.1
(0.0038) (0.0057) (0.0457) (0.0066)
DAX 0.0202 0.9480 -0.6653 0.0801 -3902.8 7813.5 7835.3
(0.0049) (0.0076) (0.0974) (0.0080)
MCL IBEX 35 0.0025 0.9957 -0.6574 0.0049 -2456.7 4921.4 4943.2
(0.0002) (0.0012) (0.0007) (0.0002)
FTSE 100 -0.0001 0.9965 -0.6022 0.0038 -2149.1 4306.2 4328.0
(0.0001) (0.0014) (0.0312) (0.0015)
DAX 0.0026 0.9942 -0.8328 0.0044 -2412.3 4832.5 4854.3
(0.0002) (0.0014) (0.0113) (0.0003)
118
Table2.12:TheempiricalestimationresultsfortheMSV
withnon-diagonalleveragemodel.Thedataisobtainedfrom
the
returnsofIBEX35,FTSE
100andDAXstockmarkets(inorder1st ,2ndand3rdseries).Theestimationisperformedvia
QMLandMCLmethods.Bollerslev-WooldridgerobuststandarderrorsareobtainedfortheQMLestimateswhilethestandard
errorsofMCLestimatesareobtainedfrom
thenumericalapproximationtotheHessian.RestrictedQMLestimationistheone
wheretherestrictionneededforMCLestimationisalsoemployedintheQMLestimationonlyforcomparisonreasons.
QML-unrestricted
fP"g 21
fP"g 31
fP"g 32
�11
�21
�31
�11
�22
�33
L11
L21
Log-like:-11179
0.8068
0.8720
0.8743
0.0147
0.0046
0.0131
0.9612
0.9548
0.9581
-0.5793
-0.1573
AIC:22400
(0.0232)(0.0201)(0.0150)(0.0015)(0.0014)(0.0017)(0.0057)(0.0065)(0.0068)(0.0284)(0.0218)
BIC:22515
L31
L22
L32
L33
fQ�g 11fQ
�g 21fQ
�g 31fQ
�g 22fQ
�g 32fQ
�g 33
-0.0522
-0.0851
-0.1835
-0.2173
0.0797
0.0788
0.0718
0.0949
0.0754
0.0770
(0.0024)(0.0027)(0.0110)(0.0124)(0.0069)(0.0034)(0.0038)(0.0058)(0.0048)(0.0081)
QML-restricted
fP"g 21
fP"g 31
fP"g 32
�11
�21
�31
�11
�22
�33
L11
L21
Log-like:-11181
0.8090
0.8730
0.8747
0.0157
0.0061
0.0161
0.9609
0.9506
0.9522
-0.6219
-0.1738
AIC:22403
(0.0226)(0.0181)(0.0161)(0.0010)(0.0028)(0.0024)(0.0050)(0.0071)(0.0083)(0.1125)(0.0144)
BIC:22518
L31
L22
L32
L33
fQ�g 11fQ
�g 21fQ
�g 31fQ
�g 22fQ
�g 32fQ
�g 33
-0.0725
-0.1245
-0.1453
-0.2143
0.0851
0.0883
0.0839
0.1101
0.0899
0.0981
(0.0143)(0.0173)(0.0299)(0.0177)(0.0041)(0.0036)(0.0077)(0.0080)(0.0067)(0.0168)
MCL
fP"g 21
fP"g 31
fP"g 32
�11
�22
�33
�11
�22
�33
L11
L21
Log-like:-4751
0.8212
0.8297
0.8542
0.0070
-0.0008
0.0043
0.9751
0.9778
0.9755
-0.6465
-0.2151
AIC:9543
(0.0001)(0.0002)(0.0001)(0.0002)(0.0001)(0.0003)(0.0002)(0.0002)(0.0004)(0.0001)(0.0001)
BIC:9658
L31
L22
L32
L33
fQ�g 11fQ
�g 21fQ
�g 31fQ
�g 22fQ
�g 32fQ
�g 33
p_val/CCMSV:0.00
-0.1052
-0.1769
-0.1387
-0.2657
0.0397
0.0347
0.0373
0.0305
0.0327
0.0351
(0.0019)(0.0195)(0.0224)(0.0377)(0.0001)(0.0001)(0.0002)(0.0001)(0.0002)(0.0001)
MCL-CCMSV
fP"g 21
fP"g 31
fP"g 32
�11
�22
�33
�11
�22
�33
L11
L21
Log-like:-5588
0.8413
0.8750
0.8835
0.0053
-0.0006
0.0030
0.9812
0.9833
09827
--
AIC:11207
(0.0001)(0.0002)(0.0002)(0.0004)(0.0002)(0.0003)(0.0003)(0.0002)(0.0005)
--
BIC:11289
L31
L22
L32
L33
fQ�g 11fQ
�g 21fQ
�g 31fQ
�g 22fQ
�g 32fQ
�g 33
--
--
0.0409
0.0339
0.0331
0.0328
0.0298
0.0327
--
--
(0.0002)(0.0002)(0.0003)(0.0002)(0.0003)(0.0003)
119
Chapter 3
Do Correlated Markets Have MoreVolatility Spillovers?
3.1 Introduction
In �nancial time series literature it is well known that even though a return series might
not have an autocorrelated structure, the squared returns tend to be serially correlated.
This is referred to as the clustering of the volatilities, which implies that periods of higher
(lower) volatility are followed by periods of higher (lower) volatility. To explain this time
varying behavior of volatilities generalized autoregressive conditional heteroskedasticity
(GARCH) models have been proposed by Engle (1982) and Bollerslev (1986). In GARCH
set up, future volatilities are modelled as a deterministic function of current volatilities
and squares of current (demeaned) returns. Later, to be able capture the e¤ects of
market interactions, multivariate GARCH (MGARCH) models have been developed.
Various MGARCH models have been well documented in Bauwens et al. (2006) and
Silvennoinen and Teräsvirta (2009).
One problem that is usually encountered in the MGARCH models is ensuring that
the conditional variance-covariance matrix is positive de�nite at any period. Bollerslev
(1990) proposed the Constant Conditional Correlation GARCH (CCC-GARCH) model
which can go around this problem. He showed that the maximum likelihood estimator
of the correlation matrix is equal to the sample correlation matrix, which by de�nition
positive de�nite. Given that the variance-covariance matrix can be decomposed to a
multiplication of the diagonal matrix of variances and the correlation matrix, the only
restriction required is that the variances are positive.
CCC-GARCH model assumes that the correlations are constant over time, which is
seen as a limitation by some papers. For example, Tse (2000) argued that the correlations
121
need not be constant and proposed a Lagrange Multiplier test for checking if the constant
correlations assumption holds. According to his results, for the spot-futures prices and
foreign exchange data the assumption of constant correlations can not be rejected while
correlations across national stock market returns are time varying. Engle (2002) and Tse
and Tsui (2002) proposed Dynamic Conditional Correlation GARCH (DCC-GARCH)
model where the correlations are changing over time. The fact that the intercept of
the equation, which drives the correlation dynamics, can be replaced by its sample
counterpart reduces the amount of parameters to be estimated signi�cantly for the DCC-
GARCH model. In the literature this replacement is referred to as correlation targeting
approach (Engle 2009). Aielli (2008) argued that estimators of DCC-GARCH model
with correlation targeting are asymptotically biased and he proposed a correction to it
which he called Corrected DCC-GARCH (cDCC-GARCH) model.
Another shortcoming of the CCC-GARCH model is that it didn�t include possible
volatility spillovers, that is to say, the future volatility of series i depends only on the
current volatilities and squared returns of series i. Jeantheau (1998) relaxed this as-
sumption in his model called Extended CCC-GARCH (ECCC-GARCH) and allowed for
volatility interactions between series. He provided the stationarity and identi�cation
assumptions and showed that the quasi-maximum likelihood (QML) estimators of this
model�s parameters are consistent. He and Teräsvirta (2004) showed that the extension
Jeantheau (1998) proposed actually allows for richer autocorrelation structure for the
squared returns compared to the CCC-GARCH model.1
In our paper, we propose a multivariate GARCH model, which we call Network
GARCH (NET-GARCH), where we allow for the volatility spillovers to occur between
the more correlated series. In this model, the future volatility of series i is in�uenced
more by the current volatility and squared return of series j if the series i and j are
highly (positively or negatively) correlated. Moreover, we let the correlations follow the
cDCC-GARCH model. Therefore at each period, the current squared returns, volatilities
and correlations are determining the future volatilities. The idea can be explained by a
real life experience that a person is more likely to smoke if a close friend of that person is
a smoker, and less likely to smoke if a casual friend is a smoker. To put it di¤erently, if a
person is a smoker, then his close friends are likely to be smokers as well while his casual
friends are less likely to be smokers. A similar behavior has been documented in Norscia
and Palagi (2011) where the authors have conducted some experiments on yawning
behavior and found out that yawning is most contagious between kin, then friends, then
1In particular, He and Teräsvirta (2004) showed that the autocorrelations of individual processes do
not necessarily decay monotonically from the �rst lag onward, as it would in the case of CCC-GARCH
model.
122
acquaintances and lastly strangers. Following this way of thought, in our model we claim
that there may be more volatility spillovers between the markets that are more related. If
an asset i is experiencing high uncertainty (high volatility), asset j that is friendly (more
correlated) with asset i is likely to import some of the uncertainty (volatility spillovers)
from asset i. To illustrate this with an example, we present Figure 3.1. To obtain this
�gure we used the �rst 40 stocks in the FTSE-100 index (alphabetically ordered)2. We
�t a VAR(1) to the squared returns vector r2t . Then for series i, we plot the coe¢ cients,
�i;j, of the squared returns of all other series j (j 6= i) along with the correlations, �ij,
between series i and j. In Figure 3.1, we report the graphs corresponding to the equations
of r27;t and r240;t. The red and blue dashed lines correspond to the means of the coe¢ cients
reported with red stars and blue circles. As we can see, most of the time the correlations
above the average (red dashed line) are matched with the coe¢ cients above the average
(blue dashed line). There are many cases where high/low correlation �ij between returns
ri;t and rj;t (points with red stars) is matched with a high/low estimated coe¢ cient �i;j of
a past squared return (points with blue circles). Some examples showing such a relation
is marked with a black straight line in the �gure. This implies that if two series i and
j have above (below) average correlation, the coe¢ cient of the previous period squared
return j (volatility spillover e¤ect) is also high (low).
In our paper, through some simulation experiments we show that the parameters
as well as the volatilities and correlations are estimated accurately. Later, we look at
the Value-at-Risk performance of the NET-GARCH model and compare it with that of
the extended model of Jeantheau (1998) with dynamic correlations (which we denote
as EcDCC-GARCH model) and with that of cDCC-GARCH model. Our results in this
section suggest that the means of the VaR estimates obtained from these models are very
similar and close to the pre-set VaR quantile of 5%. The NET-GARCH model proves to
be attractive because while it can capture the volatility spillovers with much less number
of parameters, it also can estimate the VaR�s very closely to the EcDCC-GARCH model.
In another experiment, we generate data with a BEKK and EcDCC-GARCH model
and estimate with EcDCC, NET and cDCC-GARCH models. The results obtained in
this section suggest that although NET-GARCH model overestimates the volatilities
of the series with low variance, it still does a good job in capturing the underlying
volatilities given the number of parameters in the model. Therefore while NET-GARCH
is estimating well the volatilities of an equally weighted portfolio, it is not performing
that well in estimating the volatilities of a minimum variance portfolio.
In the next section we brie�y discuss the ECCC, cDCC, EcDCC-GARCH models and
2The return data obtained from FTSE-100 index corresponds to the period 28/06/2006 - 24/01/2012.
The descriptive statistics are given in Table 1.
123
introduce the NET-GARCH setup. We give the positivity, stationarity and identi�cation
restrictions for each model. In Section 3.3, we explain the maximum likelihood estimation
of the NET-GARCH model. We use the same estimation algorithm for the other models
for comparison purposes. In Section 3.4, we report the results of our Monte Carlo
simulations. Finally Section 3.5 concludes.
3.2 Econometric Model
If we let rt to be a kx1 vector of demeaned return series, then the return equation in the
ECCC-GARCH model of Jeantheau (1998) is de�ned as:
rt = H1=2t "t, "t v N(0k; Ik)
and the conditional variance Ht is decomposed to the conditional volatilities and the
correlation matrix R:
Ht = DtRDt (3.1)
Dt = diag(h1=21t ; h
1=22t ; :::; h
1=2kt )
ht+1 = W + A"(2)t +Bht
where "(2)t = ("1t ; "2t ; :::"
kt )0. Notice that in the ECCC-GARCH model of Jeantheau
(1998), the correlations are constant, hence R is without a time subscript. If the con-
ditional variances are positive and given that R is always positive de�nite, then by
construction Ht is positive de�nite. In equation (3.1), W is a kx1 vector; A and B are
kxk matrix of parameters. If A and B are restricted to be diagonal, then the model boils
down to the CCC-GARCH of Bollerslev (1990).
Jeantheau (1998) indicated the conditions Wi > 0, Aij > 0 and Bij > 0 for i; j =
1; :::; k, are su¢ cient to for the variances to be positive. If the roots of jIk�(A+B)zj = 0lie outside of the unit circle, then the model is covariance stationary. It is noted that
when A and B are diagonal, this stationarity restriction becomes Aii + Bii < 1 for all
i. Finally, the identi�cation restrictions required for the model, among others listed in
Jeantheau (1998), are that: det(A) 6= 0, det(B) 6= 0 and A and Ik �B are coprimes.
We propose NET-GARCH model by modifying equation (3.1) as follows:
hi;t+1 = Wi + Aii"(2)it +
1Xi6=j
jpij;tj
Xi6=j
ajpij;tj"(2)jt +Biihit +1X
i6=j
jpij;tj
Xi6=j
bjpij;tjhjt (3.2)
124
where Aii andBii are the coe¢ cients of previous period squared errors and volatilities,
while a and b are the coe¢ cients of the weighted average of the previous period squared
errors and volatilities. When a and b are zeros, then there are no volatility spillovers,
and the equation (3.2) becomes the volatility equation of a CCC-GARCH model. The
weights are jpij;tjXi6=j
jpij;tjwhich are the absolute values of the correlations between series i
and the other series j at time t divided by the sum of the correlations between series
i and j, where i 6= j. Therefore at each time period, the volatility spillover between
series i and j depend on the coe¢ cients a and b and also the weights jpij;tjXi6=j
jpij;tjwhich are
proportional to the correlations between the two series. If these two series are highly
correlated, then we would expect more spillovers between their volatilities.
The process for dynamic correlations, pij;t = fRtgij, is de�ned below in (3.3), and j:jis the absolute value of the elements of its argument. The equation (3.2) can be rewritten
in multivariate form as:
ht+1 = W + A"(2)t + a �R�1jRt � Ikj"(2)t +Bht + b �R�1jRt � Ikjht
where �R is the diagonal matrix whose elements on the leading diagonal consist of
the row-sum of jRt � Ikj. The idea behind the equation (3.2) is that the spillovers
from squared returns and volatilities are multiplied by a �xed parameter and by the
weight that depends on the correlation coe¢ cient between the two series. We assume
that regardless of the positivity or negativity of the correlations between two series, the
spillovers have the same coe¢ cient. The restriction of this same coe¢ cient could be
modi�ed by including ai and bi to the equation of series i which would mean that series
i is in�uenced with the same coe¢ cient by all other series or by including aj and bj to
the equation of series i which would imply that series j in�uences all other series with
the same coe¢ cient.
For the NET-GARCH model, we use the same positivity constraints as above. The
variance stationarity should be checked inside the optimization for each t: the roots of
jIk ��(A+ a �R�1jRt � Ikj+B + b �R�1jRt � Ikj)
zj = 0 should be outside of the unit
circle. This can be done by introducing a big penalization to the likelihood when in
the iterations the parameter values imply non-stationarity. Finally, the identi�cation
restrictions can be derived similarly and should be imposed as well inside the optimization
for each t.
The correlation dynamics is de�ned according to the cDCC-GARCH model of Aielli
(2008):
125
Rt = PtQtPt (3.3)
Pt = diag(Qt)�1=2
Qt+1 = (1� �1 � �2)Q+ �1��t��0t + �2Qt
��t = diag(Qt)1=2�t:
�t = D�1t "t
where Q is replaced in the estimation by S; the sample covariance of the �t. This is
referred to as the correlation targeting approach (Engle, 2009) and it reduces signi�cantly
the number of parameters to be estimated. �1 and �2 are non-negative parameters which
satisfy �1+�2 < 1. The cDCC-GARCH model as discussed by Aielli (2008) doesn�t allow
for volatility spillovers, hence the model is de�ned by the volatility equation (3.1) where
A and B are diagonal and the correlation dynamics de�ned by (3.3). We will call the
model that consists of the volatility equation (3.1) with A and B non-diagonal and of
the correlation dynamics (3.3) by the name EcDCC-GARCH model. It should be noted
that while EcDCC and NET-GARCH models nest the cDCC-GARCH model, EcDCC
and NET-GARCH models do not nest each other.
The timing of the NET-GARCH model is as follows: at any period t, we know the
values of "t and Rt, which are functions of previous day standardized errors, and ht,
which is a function of previous day errors, volatilities and correlations. The next period
volatility ht+1, is decided depending on these three values. Similarly, the next period
correlation Rt+1 is decided depending on the standardized errors at time t, �t = D�1t "t:
3.3 Estimation
Here we describe the maximum likelihood method we use for estimating NET-GARCH
model. We need to estimate the variance parameters: � = fW 0; diagfAg; a; diagfBg; bgand the correlation parameters = f�1; �2g.The estimation procedure we use is as follows:
1. First we assume that the correlations are constant, and choose Rt = R = corr(rt).
2. Taking Rt as given, we maximize the following log-likelihood with respect to the
variance parameters, subject to positivity, stationarity and identi�cation restric-
tions for the volatilities:
L(�) = �Tk2log(2�)� 1
2
TXt=2
log jRtj �TXt=2
log jDtj �1
2
TXt=2
� 0tR�1t �t (3.4)
126
3. Taking the estimated variance parameters, �, from the �rst step estimation (step
2), we maximize the following log-likelihood with respect to the correlation para-
meters, subject to the positivity, stationarity and identi�cation restrictions for the
volatilities and correlations:
L2
� j�
�= �1
2
TXt=2
�log jRtj+ b� 0tR�1
t b�t� (3.5)
4. We take the predicted correlations, Rt, and replace Rt = Rt and repeat from step
2 on.
5. Repeat steps 2-4 until convergence. Iteration could stop depending on a norm
de�ned on the parameters, maximized value of the log-likelihood or the correlations.
The maximized value of the log-likelihood function as well as the standard errors can
be obtained from (3.4) at the convergence point. This iterative algorithm is di¤erent
than the two step estimation procedure in Engle and Shephard (2001) because in their
method there is no iteration over the variance and correlation parameters. Finally, for
comparison purposes, we estimate the EcDCC and cDCC-GARCH models with this
procedure.
3.4 Monte Carlo Experiments
3.4.1 Performance of the ML Estimator
In this section we perform a set of Monte Carlo experiments to see the small sample
performance of the ML estimator discussed in the previous section when estimating
the NET-GARCH model. In our experiments, for each parameter set, we generate 500
trivariate time series vectors of length 1000, 2000 and 5000, and estimate them the way
it is described in the previous section.
The true values of the parameters are taken from the parameter estimates obtained
from �tting a NET-GARCHmodel using the return data of the stocks listed in the FTSE-
100 index between the dates 28/06/2006 and 24/01/2012. The returns are obtained via
100 x log(Pt=Pt�1). We ordered the data alphabetically and took the FTSE-100 index
as the �rst series. The descriptive statistics for this return data is given in Table 3.1.
After eliminating the stocks for which little data is available and synchronizing the rest
of the series, we obtained 87 series of length 1356. Using this data, we estimated a
NET-GARCH model for the �rst three series (as it is also used in the Section 3.4.2), and
for three trivariate series randomly selected from the dataset.
127
We report our results in terms of MC means, standard deviations and root mean
squared errors of the estimates along with the true parameter values of the parameter set:
f�0;g = fW 0; diagfAg; a; diagfBg; b; �1; �2g. Later we plot the kernel density estimatesof the relative di¤erences between the estimated and true volatilities and correlations
calculated in the following manner for each estimator:
�bhi = 1
T
TXt=1
(bhi;t � hi;thi;t
)(3.6)
�bpij = 1
T
TXt=1
�bpij;t � pij;tpij;t
�(3.7)
where pij;t = fRtgij. We prefer to report the deviations in relative terms (dividingthe di¤erences by the true values of the volatilities or correlations) in order to have a
sense of percentage deviations from the true values.
In Table 3.2, we present the results of the MC experiments. In each of the four
experiments conducted, in the �rst row we provide the true parameter values used. Later
for the estimation results, in the �rst row we report the MC means of the estimates, in
the second row the standard deviations in parenthesis, and �nally in the third row the
root mean squared errors in brackets. In particular, the true parameter values span low
and high values of a and b parameters. In these experiments the variance parameters
are estimated quite accurately. In the second and fourth experiments, with sample size
T = 1000; the second correlation parameter �2 is estimated with a downward bias. This
could be due to the fact that the true values of a and b parameters are high. On the
other hand, when the sample size is increased, even the estimate of �2 is getting closer
to its true value.
In Figure 3.2, we plot the kernel density estimates of the relative deviations de�ned
as in (3.6) and (3.7) for the four experiments and T = 1000. In this �gure, we can
see that the relative deviations obtained for the volatility and correlation estimates are
distributed around zero. One interesting result is that when a and b parameters have
high values, the correlation parameter �2 is estimated with downward bias. (Experiment
2 and 4) However, the underlying correlations are still captured and are close to the true
correlations.
Overall, we conclude from this section that the underlying parameters as well as
the volatilities and correlations are estimated well by the maximum likelihood method
described in Section 3.3.
128
3.4.2 VaR Performance
Following Engle and Sheppard (2001) we employ a Value-at-Risk (VaR) performance
test to see the empirical validity of our model and compare its performance with EcDCC
and cDCC-GARCH models. For this purpose we construct a minimum variance portfolio
where the time varying weights to determine the portfolio return are: wt =H�1t �
Ctand
Ct = �0H�1t �; and � is a vector of ones. The portfolio variance can be then calculated
by w0tHtwt: As Engle and Sheppard (2001) points out, "...if the model is misspeci�ed,
the minimum variance portfolio should exacerbate the short coming." When estimating
the models with high number of series/parameters, it is likely that the optimization will
terminate around the starting values given to the parameters. To avoid this trap, we
consider di¤erent random starting values for the parameters, including the parameter
estimates of the cDCC-GARCH model for the series considered. Moreover, after each
optimization, we check if the iteration would continue if it was to take the estimated
values of the parameters as initial values.
When estimating the correlation parameters of DCC and cDCC-GARCHmodels with
high number of series, as Engle and Sheppard (2001) and later Engle, Sheppard and
Shephard (2008) noted, negative biases are observed, resulting in smoother correlation
estimates. For very high number of series, the estimate of the underlying correlation
is close to being constant and equal to the long-run matrix. Hafner and Reznikova
(2010) suggest the use of shrinkage methods to solve this problem. Following their
work, we make use of the shrinkage to identity method of Ledoit and Wolf (2004). In
the variance targeting approach used when estimating the correlation parameters, the
long-run covariance matrix Q is replaced by ��I :
��I = ��Ik + (1� �)S
where �Ik is the shrinkage target, � is the shrinkage intensity and � = h�; Iki =tr(�)=k is the Frobenius inner product, which coincides with the mean of the diagonal
elements of �. Hafner and Reznikova (2010) provide simulation evidence suggesting
that shrinkage to identity method is e¤ective and easy to implement. Given that we will
consider high number of series for estimating VaRs, we use this method to escape from
the possible biases of the correlation parameter estimates.
One possible doubt that could arise is that: are there volatility spillovers at all in
this data that we constructed? If there are no volatility spillovers, then it might not be
a good idea to use this data to compare the models for volatility spillovers. Therefore as
an illustration, before we perform the VaR performance tests, using the �rst three series
we estimate EcDCC-GARCH, NET-GARCH and cDCC-GARCH models and report
129
the parameter estimates and corresponding standard errors along with the values of log-
likelihood, AIC and BIC criterions. The EcDCC-GARCH estimation results for equation
(3.1):
W =
2666640:0005(0:0007)
0:0387(0:0005)
0:1051(0:0038)
377775 ; A =2666640:0617(0:0008)
0:0036(0:0004)
0:0178(0:0003)
0:0000(0:0001)
0:0455(0:0015)
0:0771(0:0017)
0:0209(0:0005)
0:0046(0:0003)
0:0679(0:0013)
377775 ; G =2666640:8838(0:0084)
0:0007(0:0009)
0:0132(0:0003)
0:0000(0:0004)
0:9349(0:0040)
0:0000(0:0002)
0:0000(0:0009)
0:0000(0:0002)
0:8367(0:0134)
377775
�1 = 0:0362(0:0004)
; �2 = 0:9249(0:0180)
; LLF = �7036:8; AIC = 14120; BIC = 14239
p_value=cDCC = 0:0000
The NET-GARCH estimation results for equation (3.2):
W =
2666640:0197(0:0003)
0:0997(0:0040)
0:0945(0:0042)
377775 ; A =2666640:0639(0:0004)
0:0613(0:0009)
0:0119(0:0004)
0:0676(0:0006)
377775 ; G =2666640:8845(0:0042)
0:9264(0:0043)
0:0000(0:0005)
0:8553(0:0073)
377775
�1 = 0:0343(0:0004)
; �2 = 0:9250(0:0110)
; LLF = �7049:3; AIC = 14125; BIC = 14192
p_value=cDCC = 0:0000
The cDCC-GARCH estimation results are:
W =
2666640:0293(0:0016)
0:1139(0:0071)
0:0776(0:0031)
377775 ; A =2666640:0864(0:0039)
0:0591(0:0035)
0:0805(0:0062)
377775 ; G =2666640:8993(0:0043)
0:9296(0:0054)
0:8844(0:0095)
377775 ;
�1 = 0:0339(0:0020)
; �2 = 0:9339(0:0125)
; LLF = �7068:3; AIC = 14159; BIC = 14216
We can see from these estimation results that some of the estimated coe¢ cients
that correspond to the volatility spillovers are statistically signi�cant at 5%. Moreover
the AIC and Likelihood Ratio tests favor the EcDCC and NET-GARCH models versus
130
the cDCC-GARCH model.3 This result implies that there are statistically signi�cant
volatility spillovers between these series in the FTSE-100 index and therefore the data
constructed can be used for our VaR performance test. Although not reported here,
similar results are obtained for more number of series.
We use the HIT test of Engle and Manganelli (2000) to test the performance of a
model�s prediction of VaR. A HIT is de�ned as a binary variable which takes value 1 if
the observed return at time t is less than the predicted Value-at-Risk: HITt = 1[rt<V aR(q)]where q is the VaR quantile. When the model predicting VaR is correctly speci�ed, the
mean of these HITs should be equal to q and independent of all the past HITs and the
predicted VaRs. Therefore an OLS regression can be constructed to test both the mean
and the independence of the HITs:
HITt � q = �0 + �1HITt�1 + �2HITt�2 + :::+ �rHITt�r + �r+1V aRt + �t (3.8)
The null hypothesis in this test is H0 : �i = 0 8 i = 0; 1; :::r+1: The test statistic andits distribution is given in Engle and Manganelli (2000). We also adopt the approach
in Engle and Sheppard (2001) where they test for the independence of HITs without
simultaneously testing if the percentage of HITs was correct. Therefore they replace q
with the mean of HITs, q, in the equation (3.8) and remove the constant in the equation:
HITt � q = �1HITt�1 + �2HITt�2 + :::+ �rHITt�r + �r+1V aRt + �t (3.9)
Using the volatility estimates, we construct one step ahead forecasts of the variance
of the minimum variance portfolio, �p; and then de�ne the 5% VaR as -1.645�p. For all
the series used in the data, the assumption of normality is rejected using a Jarque-Bera
test at 5%; mainly because of high kurtosis. Therefore this VaR level is not appropriate.
However, given that we use the test to compare EcDCC model and NET-GARCH, it
still serves our purpose.
Table 3.3 shows the VaR estimates obtained from the EcDCC, NET and cDCC-
GARCH models along with the p-values obtained for the tests performed for equations
(3.8) and (3.9). As we can see the number of parameters to be estimated increases
rapidly for EcDCC-GARCH model; while it increases linearly for the NET and cDCC-
GARCH models. The percentage of HITs are very similar across the models. In the
case of EcDCC-GARCH model, the null hypothesis that the HITs have a mean q and
3BIC reports that cDCC-GARCH model �ts the data better than the EcDCC-GARCH model. Com-
pared to AIC, BIC punishes more the number of parameters and it suggests that the increase in the
number of parameters from cDCC to EcDCC-GARCH model is not compensated by the increase in the
log-likelihood.
131
are independent of the past HITs and of the VaR estimate is rejected at 5% for 9 out
of 13 cases. When only the independence assumption is tested, in 5 of the 13 cases,
the null is rejected at 5%: In the case of NET-GARCH model, in 12 of the 13 cases the
null of the test using the equation (3.8) is rejected at 5%: On the other hand when only
the independence assumption in equation (3.9)is tested, in 9 of the 13 cases, the null is
rejected at 5%: For the cDCC-GARCH model, the null of the test using the equation
(3.8) is rejected at 5% in 12 of the 13 cases. Similarly, when the null of the test using the
equation (3.9) is rejected at 5% in 12 of the 13 cases. From these tests we conclude that
NET-GARCH model�s VaR estimates are more accurate than the cDCC-GARCH model,
and close to the estimates of the EcDCC-GARCH model. Therefore, we can say that
NET-GARCH model is doing a good job in capturing the volatility spillovers with much
less parameters than the EcDCC-GARCH model. Hence NET-GARCH model could be
preferred when high cross-sections are considered.
Table 3.4 shows the NET-GARCH estimates of the parameters related to volatility
spillovers (a and b) and of the correlation parameters (�1 and �2). As we can see, for
some of the estimations, a and b estimates are very di¤erent than zero. Moreover, the �2estimates are not necessarily small when a and b estimates are high, as opposed to the
results of Experiment 2 and 4 with T = 1000 in Section 3.4.1.
3.4.3 Robustness to Model
In this section, we provide MC evidence on the performances of EcDCC, NET and
cDCC-GARCH models when the true model is a BEKK or EcDCC-GARCH model. We
�rst estimate a BEKK-GARCH model using the �rst three, ten and thirteen series in
our dataset. Then using the parameter estimates, we generate 100 trivariate time series
vectors of length 1000 and estimate them with EcDCC, NET and cDCC-GARCHmodels.
The BEKK-GARCH model has the virtue of being very general because it allows for
spillovers between volatilities and covolatilities. Moreover, the model produces positive
de�nite covariance matrices naturally. The BEKK-GARCH(1,1,1) model is de�ned as:
Ht = CC 0 + A0"t�1"0t�1A+B0Ht�1B
where C;A and B are kxk matrices of parameters, C being lower triangular. BEKK-
GARCH(1,1,1) model is covariance stationary if and only if the eigenvalues of A A+
B B are less than one in absolute value. ( is a Kronecker product) The identi�ca-
tion restriction to eliminate observationally equivalent models is that A11, B11 and the
diagonal elements of C are positive. (See Engle and Kroner 1995 for details.)
In Figure 3.3 we report the relative di¤erences, calculated via (3.6), between the
132
volatilities estimated by each model and the true volatilities of the minimum variance
portfolio (MVP) as explained in Section 3.4.2. and equally weighted portfolio (EWP)
where the weights are equal to 1k: In all cases considered, EcDCC-GARCH outperforms
the NET and cDCC-GARCH models in estimating the volatilities. In the case of MVP,
NET-GARCH volatility estimates deviate from the true volatilities in some cases more
than the estimates of cDCC-GARCH model. However, when we look at the EWP case,
the NET-GARCH volatility estimates are closer to the true volatilities than cDCC-
GARCH volatility estimates. Finally, in the case of 8 series and 13 series, some of the
series in the data have high skewness and kurtosis, which could explain the skewness and
fat tails of the density estimates.
Next, we generate the data with EcDCC-GARCH model taking the true values of the
parameters from �tting the EcDCC-GARCH model to the �rst three, eight and thirteen
series. Then we estimate the data with EcDCC, cDCC and NET-GARCH models.
As above, in Figure 3.4 we plot the kernel density estimates of the relative di¤erences
between the estimated and true volatilities of minimum variance and equally weighted
portfolios. We obtain similar results in the sense that for the MVP case the NET-GARCH
volatility estimates deviate from the true volatilities more than the volatility estimates of
the other two models. Moreover for the EWP case the NET-GARCH volatility estimates
are closer to true volatilities than the cDCC-GARCH volatility estimates. This result
implies that even though the NET-GARCH model overestimates the volatilities of the
series with lower variance, it performs closer to EcDCC-GARCHmodel for the series with
higher variance. Moreover, the cDCC-GARCH model performs poorly in estimating the
volatilities of the series with higher variance. Finally looking at the results with the EWP,
it is remarkable that when the underlying model is an EcDCC-GARCH, the performance
of NET-GARCH in estimating the volatilities is very close to the performance of the
EcDCC-GARCH model, knowing that in the latter case there is no misspeci�cation.
Looking at Figure 3.3 and 3.4, in the case of three series we can see that the cDCC-
GARCH volatility estimates of the MVP are very similar to that of EcDCC and NET-
GARCH models while with EWP there are di¤erences. One reason could be that the
series with lower variances are receiving relatively less volatility spillovers while the
other series is receiving more. Indeed when we look at the parameter estimates of the
EcDCC-GARCHmodel reported in Section 3.4.2, we see that the unconditional standard
deviation estimates are respectively 1:3803; 3:1330 and 1:4116 and they are close to the
standard deviations reported in Table 3.1. We note that in the equation of second series
the coe¢ cient of the previous period squared residuals of the third series is 0:0771 which
is far bigger than the other spillover coe¢ cients in the estimation results. This implies
that the series with higher variance is receiving relatively more volatility spillovers.
133
We also report the root mean squared errors (RMSE) of the relative di¤erences
between the estimated and true volatilities of minimum variance portfolio and equally
weighted portfolio for each model.4 In Table 3.5, we report the RMSEs when the data is
generated by BEKK-GARCH model. In the case of three series and eight series, EcDCC-
GARCH is estimating the MVP volatilities better while in the case of thirteen series,
the RMSEs are similar across models. On the other hand in the case of EWP, EcDCC-
GARCH is estimating the volatilities better for all three simulations. NET-GARCH is
doing better in this case than the cDCC-GARCH model with three and eight series and
they are very similar with thirteen series. It is understandable that EcDCC-GARCH
captures better the volatilities generated by the BEKK-GARCH model because it is
the most general one between the three models considered. In Table 3.6, we report the
RMSEs when the data is generated by a EcDCC-GARCHmodel. It should be noted that
NET-GARCH model is estimating the volatilities better than cDCC-GARCH model in
the case of three and eight series for MVP. In the case of EWP, NET-GARCH has less
RMSE for all the simulations compared to cDCC-GARCH model. Moreover, the RMSE
of the NET-GARCH model is much closer to the RMSE of the EcDCC-GARCH model,
compared to that of the cDCC-GARCH model.
We can conclude when equal weights are given to the series in the data (EWP), on av-
erage NET-GARCH is performing quite close to the EcDCC-GARCH model. When the
underlying model is a BEKK-GARCH, with EWP the performance of NET-GARCH is
approaching to that of EcDCC-GARCH model and is at least as good as cDCC-GARCH.
When the data is generated using a EcDCC-GARCH, which is closer to a NET-GARCH
setup than BEKK-GARCH, and the portfolio is constructed with equal weights (EWP)
the NET-GARCH is doing much better than cDCC-GARCH and provides volatility
estimates close to the EcDCC-GARCH volatility estimates.
3.5 Conclusions
In this paper we propose the NET-GARCH model that captures the volatility spillovers
in a multivariate time series data while requiring the estimation of relatively less number
of parameters. The idea behind this model comes from Network Theory literature that if
a person has smoking habit, then his close friends are more likely to have the same habit
than his casual friends. Similarly, if an asset is highly volatile, then the assets closely
related to this asset are likely to import some of this high volatility.
BEKK-GARCH de�ned in Engle and Kroner (1995) and ECCC-GARCH proposed by
4Given that the mean of the squared relative di¤erences are taken, the results do not have to agree
with the Figures 3 and 4.
134
Jeantheau (1998) are two commonly used models for capturing the volatility spillovers in
the data. Even though BEKK-GARCH has a very rich structure allowing for volatilities
and covolatilities to have spillovers in-between, the parameters of this model are di¢ cult
to interpret as they do not represent directly the impact of the lagged terms on the
covariance matrix. (See Bauwens et al. 2006) ECCC-GARCH model has a simpler
structure because it uses the decomposition of the covariance matrix to volatilties and the
correlation matrix. However in both BEKK and ECCC-GARCH models the number of
parameters to be estimated increases rapidly with the number of series. Therefore �tting
these models to a time series data with a high cross sectional size becomes practically
very di¢ cult, if feasible. NET-GARCHmodel avoids this curse of dimensionality because
the number of parameters increase linearly with the number of series.
For the NET-GARCHmodel, we use the time-varying correlation dynamics of cDCC-
GARCH model proposed by Aielli (2008) as a correction to the DCC-GARCH model
of Engle (2002). Similarly, we consider the EcDCC-GARCH model where we use the
volatility set up of ECCC-GARCH model of Jeantheau (1998) but let the correlations
change following the cDCC-GARCH model.
The volatility dynamics of NET-GARCH model is similar to that of ECCC-GARCH
therefore we can adapt the positivity, stationarity and identi�cation restrictions discussed
in Jeantheau (1998). We use an iterative algorithm to estimate the NET-GARCH model
via maximum likelihood method. For comparison purposes, we estimate the EcDCC
and cDCC-GARCH models the same way. Throughout the paper, we use the return
data obtained from the stocks listed in FTSE-100 index for the period 28/06/2006 -
24/01/2012.
To show that the underlying parameters, volatilities and correlations of the data
generated by the NET-GARCH model are estimated well by the maximum likelihood
method, we perform Monte Carlo experiments where the data is generated and estimated
by this model. The true parameter values are obtained by �tting a NET-GARCH model
to the �rst three series and also to the estimation of three trivariate series randomly
selected from the data. In all four experiments, the volatility parameters of the NET-
GARCH model are estimated with small root mean squared errors. Even though with
small sample sizes, one of the correlation parameters was underestimated in two of the
four experiments, this bias disappeared with large sample sizes. The volatilities and
correlations were also estimated quite accurately.
In another section, following Engle and Sheppard (2001) we used Value-at-Risk (VaR)
performance test to check the empirical validity of the NET-GARCHmodel and compare
its performance in estimating the VaRs with that of EcDCC and cDCC-GARCH mod-
els. For this purpose we constructed minimum variance portfolios to calculate the VaR
135
estimate of each model. The results of this section suggests that NET-GARCH model
provides VaR estimates similar to that of EcDCC and cDCC-GARCH models. To test
the performance of each model in predicting correctly the VaRs, we used the test pro-
posed by Engle and Manganelli (2000). If a binary variable is de�ned which takes value
1 at time t if the observed return at time t is less than the predicted VaR, then under the
null hypothesis the mean of this binary variable should be equal to the preset VaR quan-
tile and this binary variable should be independent of all its past values and predicted
VaRs. Following Engle and Shephard (2001) we checked also only for independence of
these binary variables without simultaneously testing if the mean of this binary variable
matches the mean. The results of these tests suggest that even though NET-GARCH is
not performing as good as the EcDCC-GARCH model, it is still performing better than
the cDCC-GARCH model. Given that the number of parameters increase rapidly with
the number of series in case of EcDCC-GARCH model, NET-GARCH model seems to
be a reasonable alternative as it captures the volatility spillovers with much less number
of parameters.
We performed another set of Monte Carlo experiments to see how EcDCC, cDCC
and NET-GARCH models perform when the true data generating process is a BEKK-
GARCH or a EcDCC-GARCH model. To take the true values of the parameters, we
estimated the �rst three, eight and thirteen series by BEKK and EcDCC-GARCH mod-
els. We reported the results in this section with �gures that show the kernel density
estimates of the relative di¤erences between the volatilities estimated by each model
and the true volatilities of the minimum variance portfolios and equally weighted port-
folios. We also reported the root mean squared errors of these relative di¤erences. The
results obtained in this section suggests that NET-GARCH model is doing relatively
better than the cDCC-GARCH when the underlying model is either BEKK or EcDCC-
GARCH model in estimating the volatilities of the equally weighted portfolio. It is
remarkable that the NET-GARCH model is approximating particularly well the per-
formance of EcDCC-GARCH model with the equally weighted portfolio when the data
is generated by a EcDCC-GARCH model, given that when �tting a EcDCC-GARCH
model to this data there is no model misspeci�cation. On the other hand with the min-
imum variance portfolio NET-GARCH is not performing very well because it could be
overestimating the volatilities of the series with low variance.
Under the light of the �ndings in this paper, we can conclude that NET-GARCH
model is a reasonable alternative to the EcDCC-GARCH model to capture the volatility
spillovers. Even though with small number of series BEKK or EcDCC-GARCH models
should still be preferred because they o¤er a richer variance structure, with high number
of series the estimation of these models could become di¢ cult, when feasible. NET-
136
GARCH model proves to be useful in this case because it requires estimation of much
less parameters.
137
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139
Figure 3.1: The correlations between returns ri;t and rj;t and the coe¢ cients �i;j obtained
from regressing r2i;t on all past returns r2j;t:
0 5 10 15 20 25 30 35 400.2
0.1
0
0.1
0.2
0.3
0.4
0.5
r2j,t1
β 7,j
0 5 10 15 20 25 30 35 400.2
0.1
0
0.1
0.2
0.3
0.4
0.5
r2j,t1
β 40,j
βi,j
pij
140
Figure 3.2: Kernel density estimates of the di¤erences from the estimated volatilities
and correlations to the true ones.
0.5 0 0.50
2
4
6
8
10
∆h1,t
0.5 0 0.50
2
4
6
8
10
∆h2,t
0.5 0 0.50
2
4
6
8
10
∆h3,t
0.5 0 0.50
2
4
6
8
10
∆p12,t
0.5 0 0.50
2
4
6
8
10
∆p13,t
0.5 0 0.50
2
4
6
8
10
∆p23,t
1,2,38,51,759,17,6339,50,64zero line
141
Figure 3.3: Relative di¤erences obtained via Eq.(3.6) between the volatilities estimated
by each model and the true volatilities of the minimum variance portfolio and equally
weighted portfolio. Data generated by BEKK-GARCH.
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
8 Series
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
13 Series
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
8 Series
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8Relative to true portfolio volat ilit ies, equally weighted portfolio
3 Series
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
13 Series
0.2 0.1 0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
3 Series
Relative to true portfolio volat ilit ies, minimum variance port folio
EcDCCG ARCHcDCCGARCHNETGARCHzero line
142
Figure 3.4: Relative di¤erences obtained via Eq.(3.6) between the volatilities estimated
by each model and the true volatilities of the minimum variance portfolio and equally
weighted portfolio. Data generated by EcDCC-GARCH.
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
3 Series
Relat ive to true portfolio volat ilit ies, minimum variance portfolio
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
3 Series
Relat ive to true portfolio volat ilit ies, equally weighted portfolio
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
8 Series0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
8 Series
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
13 Series0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40
5
10
13 Series
EcDCCG ARCHcDCCG ARCHNETG ARCHzero line
143
Table 3.1: Descriptive statistics of the data constructed from FTSE-100 index and the
stocks listed in this index. Data period: 28/06/2006 - 24/01/2012Series Mean Std. Skew. Kur. Series Mean Std. Skew. Kur.
1 FTSE-100 -0.0015 1.4970 -0.0942 9.0396 45 LAND -0.0606 2.3179 -0.2188 6.68942 AAL 0.0043 3.4408 -0.1037 8.2890 46 LGEN -0.0071 3.3810 -0.3054 16.45263 ABF 0.0238 1.4689 -0.0940 6.8130 47 LLOY -0.2068 4.7186 -1.4634 26.42894 ADM 0.0229 2.4677 -1.1965 29.0575 48 MGGT 0.0281 2.3877 -0.2681 6.68505 AGK 0.1485 2.4093 0.1599 5.5519 49 MKT -0.0428 2.3019 -1.6983 24.71986 AMEC 0.0899 2.3539 -0.1948 9.2227 50 MRW 0.0233 1.6401 0.1125 6.10497 ANTO 0.0954 3.5319 0.0063 6.3748 51 NG 0.0062 1.6080 -0.1827 15.14458 ARM 0.1174 2.6773 0.0793 10.0159 52 NXT 0.0346 2.8541 0.1710 6.02059 AV -0.0365 3.4057 -1.0698 21.7725 53 OML 0.1582 6.6378 22.2919 695.289610 AZN -0.0065 1.6681 -0.1669 8.3330 54 PFC 0.1215 2.7099 -0.0196 5.410611 BA -0.0092 1.8944 -0.1849 6.0797 55 PRU 0.0247 3.4564 0.2229 11.820712 BARC -0.0797 4.3066 1.3082 27.9911 56 PSON 0.0342 1.6515 0.3752 6.114913 BATS 0.1713 3.0103 15.4167 362.1889 57 RBS -0.2271 5.3314 -6.7997 140.825614 BG 0.0555 2.3920 -0.0653 5.6275 58 RDSA 0.0353 1.8025 0.2834 9.009215 BLND -0.0482 2.4852 -0.1107 5.4603 59 REL -0.0140 1.7858 -0.6051 10.975916 BLT 0.0854 3.0561 0.1605 7.9547 60 REX -0.0278 2.0941 -0.7562 10.338717 BNZL 0.0261 1.5910 -0.2562 7.4417 61 RIO 0.0249 3.7442 -1.4811 24.182318 BP -0.0089 1.9991 -0.0299 8.2053 62 RR 0.0424 2.1727 0.0620 6.135419 BRBY 0.0745 2.7260 -0.2302 6.5810 63 RRS 0.1305 3.0205 0.2453 6.818120 BSY 0.0153 1.8452 0.3956 12.4347 64 RSA 0.0008 2.0503 0.3387 7.239921 BTA -0.0108 2.2629 -0.9214 12.9328 65 SAB 0.1119 2.2059 3.7722 55.063022 CCL -0.0043 2.4160 -0.2755 8.5126 66 SBRY -0.0150 2.0068 -1.2839 22.242323 CNA -0.0015 1.7028 -0.3791 7.1562 67 SDR 0.0367 2.8123 -0.0829 20.303924 CNE -0.0080 12.9246 0.1409 303.3931 68 SDRC 0.0259 2.8403 -0.7525 24.615525 CPG 0.0661 1.9439 -0.0919 7.1912 69 SGE 0.0199 1.8265 0.1769 5.813926 CPI 0.0092 1.5266 -0.2669 6.0986 70 SHP 0.0656 1.8443 0.2456 7.461927 CRH -0.0226 2.8603 -0.3086 7.3763 71 SMIN -0.0197 8.9780 -0.1688 599.483728 CSCG -0.0139 2.8593 1.0524 23.3745 72 SN 0.0229 1.8678 -0.0345 9.155729 DGE 0.0290 1.4093 0.2022 8.4300 73 SRP 0.0303 1.7043 -0.3114 5.776230 GFS 0.0337 1.8236 -1.6515 31.4891 74 SSE 0.0084 1.5841 -0.2191 12.343131 GKN -0.0277 3.4729 -1.3542 21.9307 75 STAN 0.0303 3.0494 0.4123 12.281032 GSK -0.0010 1.6062 0.1956 18.4226 76 SVT -0.0176 1.8385 -5.3025 115.246333 HMSO -0.0815 2.6684 -2.0512 30.6719 77 TATE -0.0013 2.2307 -2.5062 40.517834 HSBA -0.0107 2.2490 -0.7236 17.7591 78 TLW 0.0936 2.8452 0.5325 9.0712135 IAG -0.0628 3.0781 -0.1681 4.6689 79 TSCO -0.0080 1.7229 -0.2662 11.382436 IAP -0.0160 3.0994 -0.0131 12.9078 80 ULVR 0.0364 1.6135 0.0494 6.838837 IMI 0.0386 2.5659 -0.0339 5.8627 81 VED -0.0101 3.8847 -0.2963 6.381038 IMT 0.0158 1.6755 -0.2761 8.6727 82 VOD 0.0343 1.8772 -0.3156 8.463739 IPR 0.0033 2.1728 -1.1934 17.4467 83 WEIR 0.1070 2.9768 -0.2227 7.502340 ITRK 0.0853 1.9420 -0.4278 7.7864 84 WOS 0.0525 5.2736 16.9660 494.030041 ITVL -0.0246 3.0161 0.6070 9.7016 85 WPP 0.0129 2.0725 -0.2415 6.230942 JMAT 0.0330 2.3194 -0.0129 6.4140 86 WTP 0.0269 2.1872 0.1243 8.028243 KAZ 0.0021 4.2780 -0.2219 9.7506 87 XTA -0.0072 3.9689 -0.2361 7.329544 KGF 0.0087 2.4336 0.0743 4.7148
144
Table3.2:MCmeans,standarddeviationsandrootmeansquarederrorsoftheparameter.
W1
W2
W3
A11
A22
A33
aB11
B22
B33
b�1
�2
Exp.1-True
0.0197
0.0997
0.0945
0.0639
0.0613
0.0676
0.0119
0.8845
0.9264
0.8553
0.0000
0.0343
0.9250
Series:1,2,3
0.0224
0.1254
0.1169
0.0632
0.0591
0.0679
0.0118
0.8716
0.9248
0.8350
0.0032
0.0340
0.9099
T=1000
(0.0185)
(0.0652)
(0.0485)
(0.0162)
(0.0115)
(0.0173)
(0.0039)
(0.0316)
(0.0150)
(0.0417)
(0.0076)
(0.0093)
(0.0658)
[0.0187]
[0.0700]
[0.0535]
[0.0163]
[0.0118]
[0.0173]
[0.0039]
[0.0341]
[0.0151]
[0.0463]
[0.0082]
[0.0093]
[0.0675]
Exp.2-True
0.2971
0.1284
0.0047
0.0763
0.1053
0.0973
0.0045
0.8659
0.7552
0.8923
0.0171
0.0180
0.9119
Series:8,51,75
0.3882
0.1455
0.0393
0.0787
0.1041
0.0991
0.0056
0.8478
0.7402
0.8840
0.0180
0.0207
0.5509
T=1000
(0.2669)
(0.0802)
(0.0613)
(0.0243)
(0.0314)
(0.0204)
(0.0055)
(0.0572)
(0.0754)
(0.0209)
(0.0118)
(0.0141)
(0.3946)
[0.2820]
[0.0820]
[0.0704]
[0.0245]
[0.0314]
[0.0205]
[0.0056]
[0.0600]
[0.0769]
[0.0225]
[0.0119]
[0.0144]
[0.5348]
Exp.2
0.3488
0.1309
0.0212
0.0781
0.1040
0.0979
0.0045
0.8552
0.7482
0.8873
0.0191
0.0198
0.7293
Series:8,51,75
(0.1538)
(0.0590)
(0.0318)
(0.0171)
(0.0219)
(0.0153)
(0.0041)
(0.0356)
(0.0550)
(0.0156)
(0.0110)
(0.0094)
(0.3273)
T=2000
[0.1622]
[0.0591]
[0.0359]
[0.0172]
[0.0220]
[0.0153]
[0.0041]
[0.0372]
[0.0554]
[0.0164]
[0.0112]
[0.0095]
[0.3748]
Exp.2
0.3126
0.1272
0.0117
0.0765
0.1045
0.0978
0.0041
0.8627
0.7546
0.8901
0.0181
0.0185
0.8785
Series:8,51,75
(0.0690)
(0.0315)
(0.0155)
(0.0103)
(0.0137)
(0.0084)
(0.0030)
(0.0174)
(0.0332)
(0.0084)
(0.0062)
(0.0047)
(0.1466)
T=5000
[0.0708]
[0.0315]
[0.0171]
[0.0103]
[0.0137]
[0.0084]
[0.0030]
[0.0177]
[0.0332]
[0.0087]
[0.0063]
[0.0047]
[0.1503]
Exp.3-True
0.0355
0.0851
0.0858
0.1186
0.0656
0.0416
0.0074
0.8777
0.8457
0.9384
0.0058
0.0274
0.9187
Series:9,17,63
0.1183
0.1367
0.1587
0.1216
0.0734
0.0463
0.0082
0.8603
0.8109
0.9194
0.0108
0.0307
0.8079
T=1000
(0.2750)
(0.3424)
(0.2110)
(0.0361)
(0.0939)
(0.0651)
(0.0090)
(0.0767)
(0.1090)
(0.0899)
(0.0226)
(0.0450)
(0.2583)
[0.2872]
[0.3463]
[0.2233]
[0.0362]
[0.0942]
[0.0652]
[0.0090]
[0.0786]
[0.1144]
[0.0919]
[0.0232]
[0.0452]
[0.2811]
Exp.4-True
0.1563
0.2881
0.0000
0.1435
0.1412
0.0850
0.0259
0.7954
0.6537
0.8696
0.0306
0.0094
0.9697
Series:39,50,64
0.2129
0.3253
0.0319
0.1441
0.1364
0.0841
0.0270
0.7827
0.6405
0.8620
0.0317
0.0127
0.3685
T=1000
(0.1050)
(0.1227)
(0.0504)
(0.0272)
(0.0336)
(0.0207)
(0.0126)
(0.0367)
(0.0796)
(0.0275)
(0.0243)
(0.0128)
(0.4235)
[0.1193]
[0.1282]
[0.0597]
[0.0273]
[0.0339]
[0.0207]
[0.0126]
[0.0388]
[0.0806]
[0.0285]
[0.0243]
[0.0133]
[0.7354]
Exp.4
0.1787
0.3072
0.0195
0.1402
0.1373
0.0861
0.0258
0.7946
0.6499
0.8641
0.0311
0.0140
0.6399
Series:39,50,64
(0.0708)
(0.0858)
(0.0266)
(0.0212)
(0.0279)
(0.0149)
(0.0098)
(0.0297)
(0.0700)
(0.0207)
(0.0178)
(0.0091)
(0.4165)
T=2000
[0.0743]
[0.0879]
[0.0329]
[0.0215]
[0.0282]
[0.0150]
[0.0098]
[0.0297]
[0.0701]
[0.0214]
[0.0178]
[0.0101]
[0.5331]
Exp.4
0.1639
0.2962
0.0109
0.1425
0.1395
0.0848
0.0262
0.7954
0.6540
0.8681
0.0294
0.0098
0.7743
Series:39,50,64
(0.0392)
(0.0522)
(0.0149)
(0.0131)
(0.0161)
(0.0099)
(0.0066)
(0.0171)
(0.0366)
(0.0123)
(0.0114)
(0.0049)
(0.3775)
T=5000
[0.0400]
[0.0528]
[0.0184]
[0.0132]
[0.0161]
[0.0099]
[0.0066]
[0.0171]
[0.0367]
[0.0124]
[0.0115]
[0.0049]
[0.4251]
145
Table3.3:VaR
PerformanceofEcDCC,NET-GARCHandcDCC-GARCHmodels.Calculationsarebasedonaminimum
varianceportfolio.
EcDCC-GARCH
NET-GARCH
cDCC-GARCH
No.of
No.of
%of
P_values
P_values
No.of
%of
P_values
P_values
No.of
%of
P_values
P_values
series
parameters
HITs
forEq.(3.8)
forEq.(3.9)
parameters
HITs
forEq.(3.8)
forEq.(3.9)
parameters
HITs
forEq.(3.8)
forEq.(3.9)
212
0.0509
0.0809
0.0883
100.0509
0.0765
0.0571
80.0509
0.1220
0.1308
323
0.0531
0.0624
0.1224
130.0511
0.0334
0.0690
110.0524
0.0032
0.0031
438
0.0532
0.0453
0.0913
160.0546
0.0328
0.0345
140.0597
0.0040
0.0077
557
0.0553
0.0048
0.0047
190.0568
0.0481
0.0755
170.0598
0.0163
0.0295
680
0.0457
0.0521
0.0730
220.0548
0.0274
0.0435
200.0590
0.0041
0.0074
7107
0.0546
0.0209
0.0366
250.0553
0.0000
0.0001
230.0583
0.0005
0.0006
8138
0.0530
0.0282
0.0769
280.0494
0.0463
0.0714
260.0592
0.0001
0.0001
9173
0.0560
0.0256
0.0341
310.0508
0.0035
0.0044
290.0553
0.0015
0.0014
10212
0.0534
0.0937
0.0880
340.0531
0.0000
0.0001
320.0560
0.0037
0.0042
15467
0.0516
0.0271
0.0190
490.0553
0.0001
0.0001
470.0774
0.0000
0.0003
20822
0.0575
0.0000
0.0003
640.0524
0.0000
0.0000
620.0701
0.0000
0.0001
251277
0.0671
0.0035
0.0702
790.0715
0.0000
0.0000
770.0804
0.0000
0.0001
301832
0.0804
0.0000
0.1168
940.0664
0.0000
0.0003
920.0752
0.0000
0.0144
146
Table 3.4: The NET-GARCH estimates of some parametersNo. of series a b �1 �2
2 0:0056 0:0000 0:0494 0:9332
3 0:0119 0:0000 0:0343 0:9250
4 0:0134 0:0000 0:0641 0:3818
5 0:0183 0:0000 0:0411 0:4161
6 0:0215 0:0000 0:0095 0:9589
7 0:0316 0:0385 0:0306 0:6023
8 0:0344 0:0302 0:0316 0:5601
9 0:0315 0:0115 0:0169 0:8598
10 0:0330 0:0128 0:0174 0:8510
15 0:0546 0:0452 0:0192 0:7591
20 0:0246 0:0311 0:0122 0:8786
25 0:0000 0:0000 0:2379 0:1610
30 0:0002 0:0000 0:0056 0:9119
Table 3.5: The root mean squared errors of the relative di¤erences between the estimated
and true volatilities.BEKK EcDCC cDCC NET
MVP EWP MVP EWP MVP EWP
3s 0:1879 0:1527 0:2182 0:1937 0:2101 0:1706
8s 0:3464 0:2096 0:3816 0:2689 0:4067 0:2626
13s 0:3432 0:2476 0:3440 0:2985 0:3444 0:2977
Table 3.6: The root mean squared errors of the relative di¤erences between the estimated
and true volatilities.EcDCC EcDCC cDCC NET
MVP EWP MVP EWP MVP EWP
3s 0:1872 0:1315 0:2246 0:1464 0:1886 0:1388
8s 0:1445 0:0894 0:2337 0:1950 0:1661 0:1151
13s 0:1560 0:1322 0:3119 0:2225 0:3401 0:1654
147