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Chapter 8 (Brooks)

Modelling volatility and correlation

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Stylized Facts of Financial Data

The linear structural and time series models assume constant error variance. This

assumption is not realistic in financial time series data especially at high frequency. Some

of the stylized facts of financial returns data are as follows:

- Fat Tails and Leptokurtosis: Distribution of financial data has tails thicker than thenormal distribution. Tails decrease slower than . Kurtosis of financial data are almost

always greater than 3 (correspond to normal). Ignoring this may lead to under estimation

of tail risk whichhas obvious consequence for asset allocation and risk management.

- Volatility clustering: Tendency for volatility in financial markets to appear in bunches.

Thus large returns (of either sign) are expected to follow large returns and small returns

(of either sign) to follow small returns. This is due to the information arrival process.Information (news) arrive in bunches.

- Asymmetric volatility (leverage effects): Tendency of volatility to rise more following

a large price fall than following a large price rise of the same magnitude. A price fall

results in increase in firms debt to equity ratio (i.e. leverage). Shareholder's residual claim

is at larger risk.

2x

e

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A Sample Financial Asset Returns Time Series

Daily S&P 500 Returns for January 1990 December 1999

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

1/01/90 11/01/93 9/01/97

Return

Date

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KSE-100 index returns

KSE-100 Daily Returns (Jan 1990-Dec 1999)

-10

-8

-6-4

-2

02

46

810

Day

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5


0

200

400

600

800

1,000

1,200

-10 -5 0 5 10

Series: KSE index returnsSample Jan 1990 July 1999Observations 2609

Mean 0.032780Median 0.000000Maximum 12.76223Minimum -13.21425Std. Dev. 1.545728Skewness -0.265931Kurtosis 12.29654

Jarque-Bera 9425.946Probability 0.000000

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Are KSE returns different fr II

n r l returns

Simulated iid returns with same mean and standard deviation as

KSE data for Jan 1990-Dec 1999

-6

-4

-2

0

2

4

6

8

Day

IIDR

etrun

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KSE-100 index returns ( n 1 000- une 0 00 )

-10

-

--

-

0

10

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0

100

200

300

400

500

600

-8 -6 -4 -2 0 2 4 6 8

Series: KSE index returnsSample Jan 2000 July 2008Observations 2217

Mean 0.097446Median 0.095419Maximum 8.507070Minimum -8.662332Std. Dev. 1.553112Skewness -0.262416Kurtosis 6.805081

Jarque-Bera 1362.909Probabil ity 0.000000

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Dail Exchange rate ak Rs/US$

changes April 00 t ul 00

Daily Exchange Rate log change

-3

-2

-1

0

1

2

3

day

(%e

xchangeratechange)

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Dail Exchange rate ak Rs/US$

changes April 00 t ul 00

0

00

00

300

400

0

Series: Dail Exchange RateSample april 00 jul 00Observati ns 5

Mean 0.0 07Me ian 0.000000Maximum .40 344Minimum .90 03St . Dev. 0.3 5Skewness 0. 5055

urt sis 0. 9349

arque Bera 7 5. 3r babilit 0.000000

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Monthl inflation rate C I-1 ajor

cities %log change

Inflation Rate

-1.5

-1-0.5

0

0.5

1

1.5

2

2.53

M t

M

t

If

t

t

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Monthl inflation rate C I-1 ajor

cities %log change

4

6

8

1

1

14

- .5 - . .5 1. 1.5 .

eries: INFLATIONample 1995M01 006M06

Observations 1 6

Mean 0.51 396Me ian 0.471 74Maximum .366600Minimum -0.851813

t . Dev. 0.599854kewness 0.381571urtosis 3. 556 8

Jarque-Bera 3.400593robabilit 0.18 6 9

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Non-linear Models: A Definition

Campbell, Lo and MacKinlay (1997) define a non-linear data generatingprocess as one that can be written

yt=f(ut, ut-1, ut-2, )

where utis an iid error term andfis a non-linear function.

They also give a slightly more specific definition as

yt=g(ut-1, ut-2, )+ utW2(ut-1, ut-2, )wheregis a function of past error terms only and W2 is a variance term.

Models with nonlinear g() are non-linear in mean, while those withnonlinearW2() are non-linear in variance. Our traditional structural modelcould be something like:

yt=F1 +F2x2t+ ... +Fkxkt+ ut, or more compactly y = XF + u. We also assumed utb N(0,W2).

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Heteroscedasticity Revisited

An example of a structural model iswith utb N(0, ).

The assumption that the variance of the errors is constant is known as

homoskedasticity, i.e. Var (ut) = .

What if the variance of the errors is not constant?

- heteroskedasticity

- would imply that standard error estimates could be wrong.

Is the variance of the errors likely to be constant over time? Not for financial data.

Wu2

Wu2

t F1 +F2x2t+F3x3t+F4x4t+ u t

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Autoregressive Conditionally Heteroscedastic

(ARCH

) Models

So use a model which does not assume that the variance is constant.

Recall the definition of the variance ofut:

= Var(ut ut-1, ut-2,...) = E[(ut-E(ut))2 ut-1, ut-2,...]W

e usually assume that E(ut) = 0

so = Var(ut ut-1, ut-2,...) = E[ut2 ut-1, ut-2,...].

What could the current value of the variance of the errors plausiblydepend upon?

Previous squared error terms.

This leads to the autoregressive conditionally heteroscedastic modelfor the variance of the errors:

= E0 + E1 This is known as an ARCH(1) model.

Wt2

Wt2

Wt2

ut12

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Autoregressive Conditionally Heteroscedastic

(ARCH) Models (contd)

The full model would be

yt= F1 + F2x2t+ ... + Fkxkt+ ut, utb N(0, )where = E0 + E1

We can easily extend t

his to t

he general case w

here t

he error variancedepends on q lags of squared errors:

= E0 + E1 +E2 +...+Eq This is an ARCH(q) model.

Instead of calling the variance , in the literature it is usually called ht

,so the model is

yt= F1 + F2x2t+ ... + Fkxkt+ ut, utb N(0,ht)where ht= E0 + E1 +E2 +...+Eq

Wt2

Wt2

Wt2

ut12

ut q2

ut q2

W

t

2

2

1tu2

2tu

2

1tu2

2tu

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Another Way of Writing ARCH Models

For illustration, consider an ARCH(1). Instead of the above, we can

write

yt= F1 + F2x2t+ ... + Fkxkt+ ut, ut= vtWt

, vtb N(0,1)

The two are different ways of expressing exactly t

he same model.

Thefirst form is easier to understand while the second form is required for

simulation from an ARCH model.

W E Et tu! 0 1 12

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Testing for ARCH Effects

1. First, run any postulated linear regression of the form given in the equation

above, e.g. yt= F1 + F2x2t+ ... + Fkxkt+ utsaving the residuals, .

2. Then square the residuals, and regress them on q own lags to test for ARCH

of orderq, i.e. run the regression

where vtis iid.

ObtainR2 from this regression

3. The test statistic is defined as (T-q)R2 (the number of observationsmultiplied by the coefficient of multiple correlation) from the lastregression, and is distributed as a G2(q).

tu

tqtqttt vuuuu ! 22

22

2

110

2 ... KKKK

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Testing for ARCH Effects (contd)

4. The null and alternative hypotheses are

H0 : K1 = 0 and K2 = 0 and K3 = 0 and ... and Kq = 0

H1 : K1 {0 or K2 {0 orK3 {0 or ... or Kq {0.

If the value of the test statistic is greater than the critical value from the

G2 distribution, then reject the null hypothesis.

Note t

hat t

he A

RCHtest is also sometimes applied directly to returnsinstead of the residuals from Stage 1 above.

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Problems with ARCH(q) Models

How do we decide on q?

The required value ofq might be very large

Non-negativity constraints might be violated.

When we estimate an ARCH model, we require Ei>0 i=1,2,...,q

(since variance cannot be negative)

A natural extension of an ARCH(q) model which gets around some of

these problems is a GARCH model.

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Generalised ARCH (GARCH) Models

Due to Bollerslev (1986). Allows the conditional variance to be dependent

upon previous own lags

The variance equation is now

(1) This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the

variance equation.

We could also write

Substituting into (1) forWt-12 :

W

t

2

=E

0+ E

1

2

1tu +FW

t-1

2

Wt-12=E0+E1 2 2tu +FWt-22

Wt-2

2

= E0 + E12

3tu +FWt-32

Wt2

= E0 + E12

1tu +F(E0 + E12

2tu +FWt-22)

=E0+E12

1tu +E0 +E1F2

2tu +FWt-22

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Generalised ARCH (GARCH) Models (contd)

Now substituting into (2) forWt-22

An infinite number of successive substitutions would yield

So the GARCH(1,1) model can be written as an infinite order ARCH model.

We can again extend t

he GA

RCH(1,1) model to a GA

RCH(p,q):

Wt2

=E0 + E12

1tu +E0F + E1F2

2tu +F2(E0 + E1

2

3tu +FWt-32)

Wt2=E0+E1 2 1tu +E0F+E1F

2

2tu +E0F2+E1F2 2 3tu +F

3Wt-32

Wt2

=E0(1+F+F2

)+E12

1tu (1+FL+F2

L2

) +F3

Wt-32

Wt2

= E0(1+F+F2+...) + E1 2 1tu (1+FL+F2L

2+...) +FgW02

Wt2

=E0+E1 2 1tu +E22

2tu +...+Eq2

qtu +F1Wt-12+F2Wt-22+...+FpWt-p2

Wt2=

! !

q

i

p

j

jtjitiu1 1

22

0 WFEE

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Generalised ARCH (GARCH) Models (contd)

But in general a GARCH(1,1) model will be sufficient to capture the

volatility clustering in the data.

Why is GARCHbetter than ARCH?

- more parsimonious - avoids over fitting

- less likely to breech non-negativity constraints

-The ARCH part explains volatility clustering

-Th

e GARCH

part indicates the temporal dependence in volatility

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The Unconditional Variance under the GARCH

Specification

The unconditional variance ofutis given by

when

A high sum of alpha and beta would indicate volatility persistence.This would be useful in volatility forecast

is termed non-stationarity in variance

is termed intergrated GARCH

For non-stationarity in variance, the conditional variance forecasts willnot converge on their unconditional value as the horizon increases.

ar(ut) =)(1

1

0

FEE

FE 1 < 1

FE 1 u 1

FE 1 = 1

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Estimation of ARCH / GARCH Models

Since the model is no longer of the usual linear form, we cannot use

OLS.

We use another technique known as maximum likelihood.

The method works by finding the most likely values of the parameters

given the actual data.

More specifically, we form a log-likelihood function and maximise it.

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Estimation of ARCH / GARCH Models (contd)

The steps involved in actually estimating an ARCH or GARCH model

are as follows

1. Specify the appropriate equations for the mean and the variance - e.g. anAR(1)- GARCH(1,1) model:

2. Specify the log-likelihood function to maximise:

3. The software will maximise the function and give parameter values and

their standard errors

t=Q + Jyt-1+ ut , utbN(0,Wt2)Wt

2= E0 + E1

2

1tu +FWt-12

!

!

!T

t

ttt

T

t

t yyT

L1

22

1

1

2

/)(2

1)log(

2

1)2log(

2WJQWT

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Parameter Estimation using Maximum Likelihood

Consider the bivariate regression case with homoskedastic errors forsimplicity:

Assuming t

hat ut b N(0,W

2

), then yt b N( , W

2

) so th

at theprobability density function for a normally distributed random variable

with this mean and variance is given by

(1)

Successive values ofytwould trace out the familiar bell-shaped curve.

Assuming that utare iid, thenytwill also be iid.

ttt uxy ! 21 FF

tx

21 FF

!2

2

212

21

)(

2

1exp

2

1),(

W

FF

W

WFF ttttxy

xyf

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(contd)

Then the joint pdf for all the ys can be expressed as a product of theindividual density functions

(2)

Substituting into equation (2) for everyytfrom equation (1),

(3)

! !

T

t

tt

TTtT

xyxyyyf

12

2

212

2121

)(

2

1exp

)2(

1),,...,,(

W

FF

TWWFF

!

!

!

T

ttt

T

tT

Xyf

XyfXyfXyfXyyyf

1

2

21

2

421

2

2212

2

1211

2

2121

),(

),()...,(),(),,...,,(

WFF

WFFWFFWFFWFF

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(contd) The typical situation we have is that the xt andyt are given and we want to

estimate F1, F2, W2. If this is the case, then f(y) is known as the likelihoodfunction, denoted LF(F1, F2, W2), so we write

(4)

Maximum likelihood estimation involves choosing parameter values (F1,

F2,W2

) that maximise this function.

We want to differentiate (4) w.r.t. F1, F2,W2, but (4) is a product containingTterms.

! !

T

t

tt

TT

xyLF

12

2

212

21

)(

2

1exp

)2(

1),,(

W

FF

TWWFF

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Since , we can take logs of (4).

Then, using the various laws for transforming functions containinglogarithms, we obtain the log-likelihood function, LLF:

which is equivalent to

(5)

Differentiating (5) w.r.t. F1, F2,W2, we obtain

(6)

max ( ) maxlog( ( ))x x

f x f x!


(contd)

!

!

T

t

tt xyTTLLF1

2

2

21 )(

2

1)2log(

2log

W

FFTW

!

!

T

t

tt xyTTLLF

1

2

2

212 )(

2

1)2log(

2

log

2 W

FFW

!2

21

1

1.2).(

2

1

W

FF

xFx tt xyLLF

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(7)

(8

) Setting (6)-(8) to zero to minimise the functions, and putting hats above

the parameters to denote the maximum likelihood estimators,

From (6),

(9)


(contd)

!4

2

21

22

)(

2

11

2 W

FF

WxW

x tt xyTLLF

!2

21

2

.2).(

2

1

WFF

xFx ttt xxyLLF

! 0)( 21 tt xy FF

! 0

21 tt xy FF ! 0 21 tt xTy FF

! 011

21 tt xT

yT

FF

xy21

FF !

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From (7),

(10)

From (8),


(contd)

! 0)( 21 ttt xxy FF

! 0 221 tttt xxxy FF

! 0 221 tttt xxxy FF

! tttt xxyxyx )( 222 FF

! 2222 xTyxTxyx ttt FF

! yxTxyxTx ttt )( 222F

)(

222

xTx

yxTxy

t

tt

!F

! 22142 )(1

tt xy

TFF

WW

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Rearranging,

(11)

How do these formulae compare with the OLS estimators?

(9) & (10) are identical to OLS

(11) is different. The OLS estimator was

Therefore the ML estimator of the variance of the disturbances is biased,although it is consistent.

But how does this help us in estimating heteroskedastic models?

W2 21

! T

ut

W2 21

!

T

kut


(contd)

! 2212 )(1

tt xy

TFFW

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Estimation ofGARCH Models Using

Maximum Likelihood

Now we haveyt= Q + Jyt-1 + ut , utb N(0, )

Unfortunately, the LLF for a model with time-varying variances cannot bemaximised analytically, except in the simplest of cases. So a numerical

procedure is used to maximise the log-likelihood function. A potentialproblem: local optima or multimodalities in the likelihood surface.

The way we do the optimisation is:

1. Set up LLF.

2. Use regression to get initial guesses for the mean parameters.

3. Choose some initial guesses for the conditional variance parameters.

4. Specify a convergence criterion - either by criterion or by value.

Wt2

Wt2

= E0 + E12

1tu +FWt-12

! ! !

T

tttt

T

tt yy

T

L 1

22

11

2/

)(2

1

)log(2

1

)2

log(2 WJQWT

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Non-Normality and Maximum Likelihood

Recall that the conditional normality assumption forut is essential.

We can test for normality using the following representation

ut

= vtW

tv

tb N(0,1)

The sample counterpart is

Are the normal? Typically are still leptokurtic, although less so thanthe . Is this a problem? Not really, as we can use the ML with a robustvariance/covariance estimator. ML with robust standard errors is called Quasi-Maximum Likelihood or QML. The robust standard errors are developed by

Bollerslev and Wooldridge (1992)

W E E E Wt t tu! 0 1 12

2 12 v

ut

t

t

!W

t

t

t

uv

W

!

tv tv

tu

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QML Estimator

Let the normal densit for error is.

his maximization gi es the QML estimates.According to ollersle and Wooldridge 1

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Extensions to the Basic GARCH Model

Since the GARCH model was developed, a huge number of extensions

and variants have been proposed. Three of the most important

examples are EGARCH, GJR, and GARCH-M models.

Problems with GARCH(p,q) Models:

- Non-negativity constraints may still be violated

- GARCH models cannot account for leverage effects

Possible solutions: the exponential GARCH (EGARCH) model or the

GJRmodel, which are asymmetric GARCH models.

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The EGARCH Model

Suggested by Nelson (1991). The variance equation is given by

Advantages of the model

- Since we model the log(Wt

2

), th

en even if the parameters are negative, Wt

2

will be positive.

- We can account for the leverage effect: if the relationship between

volatility and returns is negative, K, will be negative.

!

TWEWKWF[W2

)log()log( 21

1

2

1

12

1

2

t

t

t

t

tt

uu

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The GJR Model

Due to Glosten, Jaganathan and Runkle

where It-1 = 1 ifut-1 < 0

= 0 otherwise

For a leverage effect, we would see K> 0.

We require E1 + K 0 and E1 u 0 for non-negativity.

Wt2 E0 E1

2

1tu +FWt-12+Kut-1

2It-1

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An Example of the use of a GJR Model

Using monthly S&P 500 returns, December 1979- June 1998

Estimating a GJRmodel, we obtain the following results.

)198.3(

172.0!ty

)772.5()999.14()437.0()372.16(

604.0498.0015.0243.1 12

1

2

1

2

1

2

! ttttt Iuu WW

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News Impact Curves

The news impact curve plots the next period volatility (ht) that would arise from various

positive and negative values ofut-1, given an estimated model.

News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR

Model Estimates:

0

0.0

0.0

0.0

0.0

0.1

0.1

0.1

-1 -0.9 -0.8 -0.7 -0.6 -0.

-0.4 -0.

-0.

-0.1 0 0.1 0.

0.

0.4 0.

0.6 0.7 0.8 0.9 1

Valu o

Lagged Shock

Valueo

Cond

iionalVariance

GARCH

G

R

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GARCH-in Mean

We expect a risk to be compensated by a higher return. So why not letthe return of a security be partly determined by its risk?

Engle, Lilien and Robins (1987) suggested the ARCH-M specification.A GARCH-M model would be

H can be interpreted as a sort of risk premium.

It is possible to combine all or some of these models together to getmore complex hybrid models - e.g. an ARMA-EGARCH(1,1)-Mmodel.

t Q+HWt-1+ut , utbN(0,Wt2)

Wt2

= E0+ E12

1tu +FWt-12

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What Use Are GARCH-type Models?

GARCH can model the volatility clustering effect since the conditionalvariance is autoregressive. Such models can be used to forecast volatility.

We could show that

Var (ytyt-1, yt-2, ...) = Var (ut ut-1, ut-2, ...)

So modelling Wt2 will give us models and forecasts forytas well.

Variance forecasts are additive over time.

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Forecasting Variances using GARCH Models

Producing conditional variance forecasts from GARCH models uses avery similar approach to producing forecasts from ARMA models.

It is again an exercise in iterating with the conditional expectationsoperator.

Consider t

he following GA

RCH(1,1) model:, utb N(0,Wt2),

What is needed is to generate are forecasts ofWT+12 ;T, WT+22 ;T, ...,

WT+s2 ;T where ;T denotes all information available up to and

including observation T.

Adding one to each of the time subscripts of the above conditionalvariance equation, and then two, and then three would yield thefollowing equations

WT+12 =E0 +E1 u2T+FWT2 , WT+22 =E0 +E1 u2T+1 +FWT+12 , WT+32 =E0 +E1

u2T+2+FWT+22

tt uy ! Q2

1

2

110

2

! ttt u FWEEW

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Forecasting Variances

usingG

ARCH

Models (Contd)

Let be the one step ahead forecast forW2 made at time T. This iseasy to calculate since, at time T, the values of all the terms on the

RHS are known.

would be obtained by taking the conditional expectation of thefirst equation at the bottom of slide 44:

Given, how is , the 2-step ahead forecast forW2 made at time T,calculated? Taking the conditional expectation of the second equation

at the bottom of slide 44:=E0 +E1E( ;T) +F

where E( ;T) is the expectation, made at time T, of , which isthe squared disturbance term.

2

,1

f

TW

2

,1fTW

2

,1

f

TW = E0 + E1

2

Tu +FWT

2

2

,1

f

TW

2

,2

f

TW

2

,2

f

TW2

1Tu2

,1

f

TW2

1Tu2

1Tu

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46

Forecasting Variances

usingG

ARCH

Models (Contd) We can write

E(uT+12 ` ;t) =WT+12

But WT+12 is not known at time T, so it is replaced with the forecast for

it, , so that the 2-step ahead forecast is given by

=E0 +E1 +F=E0 +(E1+F)

By similar arguments, the 3-step ahead forecast will be given by

=ET(E0 +E1 +FWT+22)=E0 +(E1+F)

=E0 +(E1+F)[ E0 +(E1+F) ]=E0 +E0(E1+F) + (E1+F)2

Any s-step ahead forecast (s u 2) would be produced by

2

,1

f

TW

2

,2f

TW2

,1fT

W

2

,1fT

W

2

,2

f

TW

2

,1

f

TW

2

,3

f

TW

2

,2

f

TW

2

,1f

TW2

,1

f

TW

f

T

s

s

i

if

Ts hh ,11

1

1

1

1

10, )()(

!

! FEFEE

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What Use Are Volatility Forecasts?

1. Option pricing

C = f(S, X, W2, T, rf)

2. Conditional betas

3. Dynamic hedge ratios

The Hedge Ratio - the size of the futures position to the size of the

underlying exposure, i.e. the number of futures contracts to buy or sell per

unit of the spot good.

FW

Wi tim t

m t

,,

,

!2

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48

What Use Are Volatility Forecasts? (Contd)

What is the optimal value of the hedge ratio?

Assuming that the objective ofhedging is to minimise the variance of thehedged portfolio, the optimal hedge ratio will be given by

where h = hedge ratio

p = correlation coefficient between change in spot price (S) andchange in futures price (F)

WS= standard deviation ofS

WF= standard deviation ofF

What if the standard deviations and correlation are changing over time?

Use

h p

s

F!

W

W

tF

ts

tt ph,

,

W

W

!

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49

Testing Non-linear Restrictions or

TestingH

ypotheses aboutN

on-linear Models

Usual t- and F-tests are still valid in non-linear models, but they are

not flexible enough.

There are three hypothesis testing procedures based on maximumlikelihood principles: Wald, Likelihood Ratio, Lagrange Multiplier.

Consider a single parameter, U to be estimated, Denote the MLE as

and a restricted estimate as .~U

U

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Likelihood Ratio Tests

Estimate under the null hypothesis and under the alternative.

Then compare the maximised values of the LLF.

So we estimate the unconstrained model and achieve a given maximisedvalue of the LLF, denoted Lu

Then estimate the model imposing the constraint(s) and get a new value ofthe LLF denoted Lr.

Which will be bigger?

Lre Lu comparable to RRSS u URSS

The LR test statistic is given byLR = -2(Lr - Lu) b G2(m)

where m = number of restrictions

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Likelihood Ratio Tests (contd)

Example: We estimate a GARCH model and obtain a maximised LLF of66.85. We are interested in testing whetherF = 0 i n the following equation.

yt

= Q + Jyt-1

+ ut

, ut

b N(0, )= E0 + E1 + F

We estimate the model imposing the restriction and observe the maximisedLLF falls to 64.54. Can we accept the restriction?

LR = -2(64.54-66.85) = 4.62.

The test follows a G2(1) = 3.84 at 5%, so reject the null.

Wt2

Wt2 ut1

2 2

1tW

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52

Modelling Volatilit of KSE Returns

Lets consider dail KSE-100 index return data from an 1 000-une 0 008. We hold last obser ations forout of sample

forecast comparison. Here is the ofplot returns hich shows t picalpattern of financial data

KSE-100 index returns an 1 000 une 0 008)

-10

-8

--

-

0

8

10

ay

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53

Identifying GARCH order

Correlogramof KSE-return here seem tobe some significant

autocorrelations up toand including10 lags. he mean e uation mayin ol e toomanyparameters

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Identifying GARCH order

Correlogramof S uared KSE-return AC of first four lags and at some higher large are large. ossibly

indicating that simple ARCH may in ol e toomanyparameters.

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Testing forARCH Effects in Returns

ARCH LM test can be directlyapplied tokse-returns to see if s uared returnshas autoregressi e structure

LM(11 =TR = 07*0.1484= 7. al=@chis (327. ,11 =0.000

As suggested in the literature e.g. Enders (2004,p146 we start with the mostparsimonious model for conditional olatility that has often been found tobesatisfactory in de eloped stockmarkets i.e. GARCH(1,1 ,.

If it f ails to satisfyany diagnostics,more complicated model are needed.

return

ss

quaredo

flags

oftscoefficienarewhereHo kk HHHH 0...: 21 !!!!

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56

Tentati e Model ARMA(1,1 -GARCH

(1,1 Lets examine the fit ofARMA(1,1 -

GARCH(1,1 model.All the coefficients inmean and olatility e uations are significant.Also the GARCH model seem tobe

covariance stationaryas the sumofalpha+beta =0.18 2+0.7 25=0.937

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57

Diagnostics rdinaryand StandardizedResiduals

The estimated models residualsshould be uncorrelated and the

residuals should not contain anyremaining conditional volatility.

Kurtosis of standardizedresiduals (residuals divided byconditional standard deviationis smaller than the kurtosis ofordinary residuals,which shouldbe the case if GARCH is asuitable model

- - -

-

Series !

rdinary R " SIds

Sample # $ $

#

%

!

bservations$ $

#

&

' ean -(

.( )

&%

#

)

Median -(

.(

#

$ % 0 $

Ma1

imum%

.% 2 2 3 3 &

Minimum -%

.% 0 ) & 3 $

Std.Dev.#

.2

4&

%

#

$

Skewness -(

.#

0 3

#

2

#

Kurtosis&

.( ) % 3 0 3

Jar5

ue-6

era #

2

#

&

.2

# #

Probability (

.( ( ( ( ( (

7

8 7 7

9 7 7

@ 7 7

A

7 7

B 7 7

C 7 7

-C -A

-9 7 9A

SeriesStandardized R

"SIDs

Sample#

$ $

#

%

!

bservations$ $

#

&

Mean -(

.( $ 3

2

#

4

Median -(

.(

# #

( 3 %

Ma1 imum

4

.$

#

2 % ) %

Minimum -0

.)

#

0 4 4 3

Std.Dev.(

.3 3 3 3 $ 3

Skewness -(

.2

#

( 0 4 (

Kurtosis2

.0

&3 ( 3 )

Jar5

ue-6

era & 2

3

.)

&) %

Probability (

.( ( ( ( ( (

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58

Model Diagnostics Standardized

Residuals Standardized residuals and

s uared standardizedresiduals do not give

evidence ofanymisspecification

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59

Testing forLeverage Effects (Enders,

p-148 If there is no leverage, the regression of s uared standardized

residuals on their level should give an overall test which isinsignificant and individual t-values insignificant.

sresid(-1 and sresid(-2)are significant as well as

overall -test.

This indicates presence of

leverage effect. Conditional

variance ofkse returns is

affected more by

negative shocks.

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EGARCH Estimates

The coefficient of terminvolving lag residual allowsthe sign of residual toaffect

conditional variance. Ifasymmetry is present, thiscoefficient (in this case c(6))should be negative andsignificant. If this term is zerothere is noasymmetry. In

this case the term is negativeand significant whichconfirms previous diagnostictest.

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EGARCH model

The estimated EGARCH model gives

Apositive shock in error last period (good news)ofone unitincreases volatilityby =-0.1103 +0.3173=0.207 units

A negative shock in error last period (bad news)ofone unitincreases volatilityby =-(-0.1103) +0.3173=0.4276 units

which is more than double hence volatility responds to

shocks in asymmetric way.

2

1

1

1

1

12 log8997.01103.03173.0173.0log

! tt

t

t

tt

uuW

WW

W

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ARCH-in Mean

Using either standarddeviation orvariance inthe mean e uation gives

insignificant estimate ofriskpremium.Thus in thiscase ARCH-M is not asatisfactory description of

kse-returns.

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Model Selection

The following table indicates that log likelihood correspondsto EGARCH (1,1)(with T distributed errors) is themaximum among competing models. Compared to normal

case parameter estimates in the case of EGARCH with terrors are slightly different.Akaike and Schwarz criteriaalso give similar results.

Model Log-Li eli ood

GARCH(1,1) -Normal Errors -3792.053

GARCH(1,1) -T Errors -3690.913

GARCH(1,1) -GED Errors -3685.821

EGARCH(1,1) Normal Errors -3787.982EGARCH(1,1) T Errors -3677.53

EGARCH(1,1) GED Errors -3678.178

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Estimated Volatility

Volatility estimates from EGARCH model resembles more closely to realizeds uared returns

0

1 0

2 0

3 0

4 0

5 0

6 0

2 5 5 0 7 5 1 0 0

V G A R C H T V E G A R C H T K S E ^2

D a t e

volatility

E s t im a t e d V o la t i l ity a n d S q u a re d K S E R e t u rn s

ch8_ slides 21 july 2010

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