forecasting in financial market using markov regime...

OutlineMotivation

Research ObjectivesMethodology

Scope and limitationsResearch Methodology

Descriptive of model

FORECASTING IN FINANCIAL MARKET USINGMARKOV REGIME SWITCHING ANDPRINCIPAL COMPONENT ANALYSIS.

NOP SOPIPAN

September 13, 2012

NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.

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1 Motivation

2 Research Objectives

3 Methodology

4 Scope and limitations

5 Research MethodologyHeteroskedasticity ModelSpurious high persistence problemMulticollinearily

6 Descriptive of model


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In Thai

การพยากรณในตลาดทางการเงน ดวย

วธการวเคราะหองคประกอบหลก และ วธสบเปลยนสถานะมารคอฟ


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The Beginning

Prof.Dr.Pairote Sa-ayatham Prof.Dr.Bhusana Premanode


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Characteristics of returns


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Property in Financial returns

(Weakly) Stationary process.

Look liked White noise process

The model assume random walk model.


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Forecast Financial Return

Let Pt denote the series of the financial price at time t andrtt>0 be the logarithmic return (in percent) on the financialprice at time t be a sequence of random variables on a probabilityspace(Ω,F,P) , i.e.

rt = 100 · ln(Pt

Pt−1) (1)

Let rt be explain in mean equation, i.e.

rt = µt + εt (2)

with mean and variance:

µt = E (rt |Ft−1), ht = Var(rt |Ft−1) = E [(rt − µt)2|Ft−1]


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Forecasted inFinancial Return

Mehmet A.(2008), Marcucci J. (2005) assume mean equationin returns is constant

Easton and Faff (1994), and Kyimaz and Berument (2001)assume mean equation in returns with a one week delay intothe regression model

Supot C.(2003)assume mean equation in returns withAutoregressive process.


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Construct Model

The constant mean equation cannot be forecast accuratelydue to the inaccuracy of the financial data, since the financialreturns depend concurrently and dynamically on manyeconomic and financial variables.

The fact that the return has a statistically significantautocorrelation indicates that lagged returns might be usefulin predicting future returns.

We consider add some explanatory variables and stationaryAutoregressive Moving-average order p and q (ARMA (p,q)).Then we add them into a mean equation in order to increaseaccuracy.


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Construct Model


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Research Objectives

The objectives of the research are the following :

1 Construct model of equation (3) for forecasting of financialreturn.

2 Solve multicollinearity problem in explanatory variables ofequation (3) by using Principal Component Analysis (PCA).

3 Construct heteroskedasticity model for forecasting of volatilitywith GARCH-type model and Markov Regime SwitchingGARCH model.

4 Compare and evaluate the performance of models betweenconstant mean equation and model of equation (3) with lossfunctions.


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Research Objectives

The process of research had been following :

1stPaper.N. Sopipan, P. Sattayatham and B. Premanode ,”Forecasting Volatility of Gold Price using Markov RegimeSwitching and Trading Strategy”, Journal of mathematicalfinance.Vol. 2(1), pp 121-131 , 2012.

2ndPaper. P. Sattayatham, N. Sopipan and B. Premanode(2012), ”Forecasting the Stock Exchange of Thailand usingDay of the Week Effect and Markov Regime SwitchingGARCH.”, Journal of Forecasting. (in progress)

Next Step Use PCA to solve multicollinearity problem inexplanatory varibles in mean equation forecast return.


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Scope and limitations

The forecasting in Markov Regime Switching for simplicity,this thesis assume presence of two regimes and order of theGARCH-type is (1,1).

The standardized errors follow student-t, generalized errordistributions (GED) and normal distribution.


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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily

Heteroskedasticity


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Heteroskedasticity

The generalized autoregressive conditional heteroskedasticity(GARCH)-type models mainly capture for problem ofheteroskedasticity.

GARCH model

The Exponential GARCH (EGARCH) model

GJR-GARCH model


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Heteroskedasticity model

GARCH(1,1):[Bollerslev (1986)]

εt = ηt√

ht , ht = α0 + α1ε2t−1 + β1ht−1 (3)

EGARCH(1,1):[Nelson (1991)]

ln(ht) = α0 + α1|εt−1√ht−1|+ β1ln(ht−1) + ξ

εt−1√ht−1

(4)

GJR-GARCH(1,1):[Glosten, Jagannathan and Runkle(1993)]

ht = α0+α1ε2t−1(1−Iεt−1>0)+β1ht−1+ξε2t−1(Iεt−1>0) (5)


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Spurious high persistence problem

0 200 400 600 800 1000 12000

20

40

60

80

100

120

140

Vola

tilit

y


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Review of Markov Regime Swithcing

Hamilton (1989,1990) extended MRS for analyzing structuralbreaks in financial return.

Hamilton and Susmel (1994) and Cai (1994) proposed MRSARCH.

Gray (1996)proposed MRS GARCH.

Klassen (2002) modification of Gray’s MRS GARCH.


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Concept of Markov Regime Swithcing

Generalization of the simple dummy variables approach.

Allow regimes (called states) to occur several periods overtime.

In each period t the state is denoted by St .

There can be N possible states: St = 1, . . . ,N.


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Markov Regime Swithcing

Probability St equals some particular value j depends on thepast only through the most recent value St−1:PSt = j |St−1 = i , St−2 = k , . . . = PSt = j |St−1 = i =pij .

Process is described as an N-state Markov Chain withtransition probabilitiespiji ,j=1,2,...,N and pi1 + pi2 + . . .+ piN = 1


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The conditional probability density function for theobservations rt given the state variables St ,St−1 and Ft−1 isf (rt |St , St−1,Ft−1)

The chain rule for conditional probabilities yields then for thejoint probability density function for the variables rt ,St ,St−1given Ft−1 isf (rt ,St ,St−1|Ft−1) = f (rt |St ,St−1,Ft−1)Pr(St ,St−1|Ft−1)

The log-likelihood function to be maximized with respect tothe unknown parameters becomesl(θ) =

∑Tt=1[log(

∑NSt=1

∑NSt−1=1)f (rt ,St ,St−1|Ft−1)]


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The conditional probability density function for theobservations rt given the state variables St ,St−1 and Ft−1 isf (rt |St , St−1,Ft−1)

The chain rule for conditional probabilities yields then for thejoint probability density function for the variables rt , St , St−1given Ft−1 isf (rt ,St ,St−1|Ft−1) = f (rt |St ,St−1,Ft−1)Pr(St ,St−1|Ft−1)

The log-likelihood function to be maximized with respect tothe unknown parameters becomesl(θ) =

∑Tt=1[log(

∑NSt=1

∑NSt−1=1)f (rt ,St ,St−1|Ft−1)]


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Markov Regime Swithcing GARCH models

The MRS-GARCH model with only two regimes can be representedas follows.

rt = µt,St + εt = µt,St + ηt√

ht,St (6)

ht,St = α0,St + α1,Stε2t−1 + β1,St ht−1 (7)

where St = 1, 2 , µt,St is the mean and ht,St is the volatility underregime St on Ft−1 .


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Multicollinearily

Multicollinearity is a statistical phenomenon in which two or morepredictor variables in a multiple regression model are highlycorrelated.


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Multicollinearily

For the mean equation in (3) i.e.

rt = µt + εt ,

µt = φ0 +k∑

i=1

βiXit +

p∑i=1

φi rt−i −q∑

i=1

θiεt−i

Multicollinearity, or high correlation between the explanatoryvariables in a regression equation. One method for removing suchmulticollinearity and redundant independent variables is principalcomponent analysis (PCA).


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Principal Component Analysis

The idea of PCA is to find linear combinations αi such that Zi andZj are uncorrelated for i 6= j and the variances of Zi are as large aspossible. More specifically:

The first principal component of X is the linear combinationZ1 = α

′1X that maximizes Var (Z1) subject to the constraint

α′1α1 = 1.

The second principal component of X is the linearcombination Z2 = α

′2X that maximizes Var (Z2) subject to

the constraint α′2α2 = 1 and Cov(Z2,Z1) = 0.


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Principal Component Analysis (Cont.)

The kth principal component of X is the linear combinationZk = α

′kX that maximizes Var (Zk) subject to the constraint

α′kαk = 1 and Cov(Zk ,Zk ′) = 0 for k 6= k ′


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Principal Component Analysis (Cont.)

The new variables from the PCA become ideal to use as predictorsin a regression equation since they optimize spatial patterns andremove possible complications caused by multicollinearity.


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Model of returns

The model of returns (2) as follow:

rt = µt + εt

This thesis use three mean equations of return as follow:(1) Mean equation are constants.

µt = δ.

µt = δS(t).


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Model of returns

(2) Mean equation as stationary ARMA(p,q) includes day ofthe week effect.

µt = Σ5i=1βiDit + Σp

j=1φj rt−j − Σqk=1θkεt−k ,

whereDit , i = 1, ..., 5 are dummy variables which take on the value of 1 ifthe corresponding return of the five days respectively and 0otherwise,βi , i = 1, ..., 5 are coefficients which represent the average returnfor each day of the week ,φj , θk , i = 1, ..., p, k = 1, ..., q, are coefficients which represent theARMA (p, q).


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THANK YOU FOR ATTENTION


forecasting in financial market using markov regime...

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