# Performance of GARCH models in forecasting stock market volatility

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Performance of GARCH Models inForecasting Stock Market Volatility

CHOO WEI CHONG,* MUHAMMAD IDREES AHMADand MAT YUSOFF ABDULLAH

Universiti Putra Malaysia, Malaysia

ABSTRACT

This paper studies the performance of GARCH model and its modifica-tions, using the rate of returns from the daily stock market indices of theKuala Lumpur Stock Exchange (KLSE) including Composite Index, TinsIndex, Plantations Index, Properties Index, and Finance Index. The modelsare stationary GARCH, unconstrained GARCH, non-negative GARCH,GARCH-M, exponential GARCH and integrated GARCH. The para-meters of these models and variance processes are estimated jointly usingthe maximum likelihood method. The performance of the within-sampleestimation is diagnosed using several goodness-of-fit statistics. We observedthat, among the models, even though exponential GARCH is not the bestmodel in the goodness-of-fit statistics, it performs best in describing theoften-observed skewness in stock market indices and in out-of-sample (one-step-ahead) forecasting. The integrated GARCH, on the other hand, is thepoorest model in both respects. Copyright# 1999 John Wiley & Sons, Ltd.

KEY WORDS forecasting volatility; GARCH; time-series; rate of returns

INTRODUCTION

Following the seminal work of Mandelbrot (1963) and Fama (1965), many researchers havefound that the empirical distribution of stock returns is significantly non-normal, such as Hsuet al. (1974), Hagerman (1978), Lau et al. (1990), Kim and Kon (1994), etc. They found that(1) the kurtosis of the stock returns time series is obviously larger than the kurtosis of the normaldistribution, in order words, the time series of stock returns are leptokurtic; (2) the distribution ofstock returns is skewed, either to the right (positive skewness) or to the left (negative skewness);(3) the variance of the stock returns is not constant over time or the volatility is clustering. Someresearchers regarded this as the persistency of the stock market volatility and the financial analystcalled this uncertainty or risk. This uncertainty is crucially important in modern finance theory.Before the seminal paper by Engle (1982), the uncertainty of speculative prices, changing

CCC 02776693/99/05033311$17.50 Received January 1998Copyright # 1999 John Wiley & Sons, Ltd. Accepted December 1998

Journal of Forecasting

J. Forecast. 18, 333343 (1999)

* Correspondence to: Choo Wei Chong, Department of Management and Marketing, Faculty of Economics andManagement, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia. Email: wcchoo@econ.upm.edu.my

over time (Mandelbrot, 1963; Fama, 1965) measured by the variances and covariance has beenaccepted for decades.Many conventional time series and econometric models work only if the variance is constant.

Until lately, the financial and economic researchers have started modelling time variationin second- or higher-order moments. Engle (1982) has characterized the changing variancesusing the Autoregressive Conditional Heteroscedasticity (ARCH) model and its extensions andmodifications. Since then, hundreds of researchers have applied these models to financial timeseries data.In many of the applications, the linear ARCH(q) model requires a long lag length of q. The

alternative and more flexible lag structure is the generalized ARCH (GARCH) introduced byBollerslev (1986). It is proven that a small lag such as GARCH(1,1) is sucient to model thevariance changing over long sample periods (French et al., 1987; Franses and Van Dijk, 1996).Even though GARCH model can eectively remove the excess kurtosis in returns, the

GARCHmodel cannot cope with the skewness of the distribution of returns, especially the stockmarket indices which are commonly skewed. Hence, the forecasts and forecast error variancesfrom a GARCH model can be expected to be biased for a skewed time series. Recently, a fewmodifications to the GARCH model have been proposed which explicitly take skewed distri-butions into account. One of the alternatives of non-linear models that can cope with skewnessis the Exponential GARCH or EGARCHmodel introduced by Nelson (1990). For stock indices,Nelsons exponential GARCH is proven to be the best model of the conditional hetero-scedasticity. Franses and Van Dijk (1996) found that there are diculties when comparing thisEGARCHmodel to out-of-sample forecasting performance. In order to know the out-of-sampleforecasting performance of EGARCH, we compare the performance of EGARCH and the otherfive modifications of the GARCH model to the simple random walk forecasting scheme.The models are presented in the following section. The third section gives the background to

stock market indices data and the methodology used in this study. All the results will be discussedin the fourth section. Conclusions are presented in the final section.

MODELS

Consider a stock market index It and its rate of return rt , which we construct asrt It It1=It1. The index t denotes daily closing observations. Instead of return, rate ofreturn was used to make the data dimension less. The conditional distribution of the series ofdisturbances which follows the GARCH process can be written as

etjCt1 N0; ht

where Ct1 denotes all available information at time t 1. The conditional variance ht is

ht w Xqi1

aie2t1

Xpj1

bjhtj

where p5 0, q4 0 and w4 0, ai5 0, bj5 0 for non-negative GARCH (NG(p,q)) process. TheGARCH(p,q) model reduces to the ARCH(q) process when p 0 and at least one of the ARCH

Copyright # 1999 John Wiley & Sons, Ltd. J. Forecast. 18, 333343 (1999)

334 Choo Wei Chong et al.

parameters must be non-zero (q4 0). The GARCH regression model for the series of rt can bewritten as

fsBrt m et; with fsB 1 f1B fsBs

et

hp

tetet N0; 1

ht w Xqi1

aie2t1

Xpj1

bjhtj

where B is the backward shift operator defined by Bkyt ytk. The parameter m reflects aconstant term, which in practice is typically estimated to be close or equal to zero. The order of sis usually 0 or small, indicating that there are usually no opportunities to forecast rt from its ownpast. In other words, there is never an auto-regressive process in rt . The parameter of w, ai and bjcan be unconstrained, thus yielding the unconstrained GARCH (UG(p,q)) model. If theparameters are constrained such that

Xqi1

ai Xpj1

j5 1 ;

they imply the weakly stationary GARCH (SG(p,q)) process since the mean, variance, andautocovariance are finite and constant over time. Sometimes the multistep forecasts of thevariance do not approach the unconditional variance when the model is integrated in variance;that is,

Xqi1

ai Xpj1

bj 1

The unconditional variance for the IGARCHmodel does not exist. However, it is interesting thatthe integrated GARCH or IGARCH (IG(p,q)) model can be strongly stationary even though it isnot weakly stationary (Nelson, 1990a,b). The exponential GARCH or EGARCH (EG(p,q))model was proposed by Nelson (1991). Nelson and Cao (1992) argue that the non-negativityconstraints in the linear GARCHmodel are too restrictive. The GARCHmodel imposes the non-negative constraints on the parameters, ai and bj , while there is no restriction on these parametersin the EGARCH model. In the EGARCH model, the conditional variance, ht , is an asymmetricfunction of lagged disturbances, eti :

lnht w Xqi1

aigZti Xpj1

bjlnhtj

where

gZt yZt gjZtj EjZtjZt etj

hp

t

Copyright # 1999 John Wiley & Sons, Ltd. J. Forecast. 18, 333343 (1999)

GARCH Models and Stock Market Volatility 335

The coecient of the second term in g(Zt) is set to be 1 (g 1) in this formulation. Note thatE jZt j (2/p)1/2 if Zt N0; 1. The GARCH-M ((G(p,q)-M) model has the added regressorthat is the conditional standard deviation

rt m d

hp

t etet

hp

tet

where ht follows the GARCH process.In the case of stock market indices, the GARCH models are often preferred for analysis.

Furthermore, since a small lag of the GARCH model is sucient to model the long-memoryprocesses of changing variance (French et al., 1987; Franses and Van Dijk, 1996), theperformance of GARCH models is evaluated by using SG(1,1), UG(1,1), NG(1,1), G(1,1)-M,EG(1,1) and IG(1,1) in this study.

DATA AND METHODOLOGY

The data used in this paper is daily observed stock market indices in the Kuala Lumpur StockExchanges (KLSE), including Composite Index, Tins Index, Plantations Index, Properties Indexand Finance Index. This data was collected from 1 January 1989 to 31 December 1990. In thisstudy, we consider the daily closing prices as the daily observations.Some characteristics of the rate of returns, rt, are given in Table I. The number of observations

are 483 for all the five indices. The means and variances are quite small. The high kurtosisindicates the necessity of fat-tailed distributions to describe these variables. The estimatedmeasure of skewness is either positive or negative and is large in an absolute sense.The family of GARCH models are estimated using the maximum likelihood method. This

enables the rate of return and variance processes to be estimated jointly. The log-likelihoodfunction is computed from the product of all conditional densities of the prediction error:

l XNt1

1

2ln2p lnht

e2tht

where et rt m and ht is the conditional variance. When the GARCH(p,q)-M model isestimated, et rt m d

hp

t. When there are no regressors (trend or constant, m), the residualset are denoted as rt or rt d

hp

t. The likelihood function is maximized via the dual quasi-Newton

Table I. Summary statistics of data on rate of returns from 1989 to 1990

Indices n Mean (104) Variance (104) Skewness KurtosisComposite 483 7.613 1.772 1.533 18.772Tins 483 1.733 3.278 0.070 8.630Plantations 483 0.698 1.660 0.019 16.593Properties 483 7.903 5.367 1.783 26.246Finance 483 6.868 2.988 0.303 13.142

Source of data: Investors Digest, KLSE.

Copyright # 1999 John Wiley & Sons, Ltd. J. Forecast. 18, 333343 (1999)

336 Choo Wei Chong et al.

and trust region algorithm. The starting values for the regression parameters m are obtained fromthe OLS estimates. When there are autoregressive parameters in the model, the initial values are

obtained from the YuleWalker estimates.

In order to test for independence of the indices series, the portmanteau test statistic based onsquared residual is used (McLeod and Li, 1983). This Q-statistic is used to test the non-lineareects, such as GARCH eects, present in the residuals. The GARCH(p,q) process can beconsidered as an ARMA(max(p,q),p) process. Therefore, the Q-statistic calculated from thesquared residuals can be used to identify the order of the GARCH process. The Lagrangemultiplier (LM) test, proposed by Engle (1982), is used to test for the ARCH disturbances.The LM and Q-statistics are computed from the OLS residuals, assuming that disturbances is

white noise. The Q and LM statistics have an approximate (w2q) distribution under the white-noise null hypotheses.Various goodness-of-fit statistics are used to compare the six models in this study. The

diagnostics are the mean of square error (MSE), the loglikelihood (Log L), Schwarzs Bayesianinformation criterion (SBC) by Schwarz (1978) and Akaikes information criterion (AIC) (Judgeet al., 1985).The true volatility is measured to evaluate the performance of the six GARCH models in

forecasting the volatility in stock returns. As in the studies by Pagan and Schwert (1990) and Dayand Lewis (1992), the volatility is measured by

vt rt r2

where r is the average return. The measure of the one-step-ahead forecast error is

et1 vt1 h^t1where ht1 is generated using the ht equations of the GARCH models being studied. Theestimated parameters of the GARCH models such as w, a, b, y and d are substituted during thegeneration of ht1 . In order to show the performance of GARCH models over a nave no-changeforecast, the forecast errors of the random walk (RW) is calculated as follows:

et1 vt1 vtThis is a very important nave benchmark in comparison of the forecasts from the GARCHmodels (Brooks, 1997).

RESULTS AND DISCUSSION

The parameter estimates for the six variation GARCH models used in this study is presented inTable II(a). The additional parameter estimates of EG(1,1) and G(1,1)-M are reported inTable II(b). These within-sample estimation results enable us to know the possible usefulness ofthe GARCH models in modelling the volatility of stock returns. It is clear from Tables II(a) andII(b) that almost all the parameter estimates including m, o, a, b, y and d are significant at the 5%level. Hence, the constant variance model can be rejected, at least within sample. For linearGARCH models such as SG(1,1), the sum of the a and b parameters is close to unity for theFinance Index. However, for the Properties Index, we observed that a b 1 for a stationary

Copyright # 1999 John Wiley & Sons, Ltd. J. Forecast. 18, 333343 (1999)

GARCH Models and Stock Market Volatility 337

Table II(a). Estimation results of rate of returns for five stock market indices

Index ModelParameter estimates

m (104) t-ratio o 105) t-ratio a t-ratio b t-ratioComposite SG(1,1) 3.229 0.523 8.11768 6.500 0.420 2.938 0.179 1.722

UG(1,1) 3.228 0.512 8.11766 6.525 0.420 2.911 0.179 1.727NG(1,1) 3.229 0.514 8.11763 6.519 0.420 2.893 0.179 1.718G(1,1)-M 53.301 2.384 7.058 6.686 0.488 3.066 0.212 2.249EG(1,1) 3.7492 2.815 5.237 0.721 4.691 0.481 4.935

448,246.11IG(1,1) 4.593 0.888 7.299 7.599 0.922 15.67 0.078

Tins SG(1,1) 8.543 1.101 1.330 5.620 0.317 3.485 0.310 3.367UG(1,1) 8.5474 1.108 1.330 5.888 0.317 3.605 0.310 3.457NG(1,1) 8.543 1.101 1.330 5.620 0.317 3.486 0.310 3.370G(1,1)-M 86.971 2.129 1.344 5.604 0.322 3.704 0.296 3.289EG(1,1) 16.459 7.381 32,767.92 4.971 0.485 5.117 0.591 7.317IG(1,1) 15.220 2.159 1.129 6.157 0.755 10.75 0.245

Plantations SG(1,1) 1.434 2.578 0.476 6.086 0.458 2.782 0.324 3.682UG(1,1) 1.434 2.520 0.476 5.900 0.458 2.828 0.324 3.741NG(1,1) 1.434 2.541 0.476 6.213 0.458 3.123 0.324 3.999G(1,1)-M 5.574 3.575 0.458 5.417 0.548 3.222 0.290 3.763EG(1,1) 1.648 6.516 28,679.48 5.378 0.731 6.098 0.671 11.16IG(1,1) 1.836 4.100 0.424 6.223 0.730 11.13 0.270

Properties SG(1,1) 5.404 0.884 0.264 4.717 0.435 10.39 0.565 UG(1,1) 2.427 0.413 0.170 3.099 0.652 5.834 0.545 12.38NG(1,1) 2.429 0.417 0.170 2.977 0.652 6.155 0.545 12.25G(1,1)-M 3.503 3.258 0.171 3.006 0.663 6.006 0.527 11.52EG(1,1) 1.003 0.169 8034.68 4.180 0.785 8.839 0.892 38.81IG(1,1) 5.403 0.884 0.264 4.717 0.435 10.39 0.565

Finance SG(1,1) 9.229 1.385 0.402 2.937 0.262 3.733 0.631 10.15UG(1,1) 9.229 1.347 0.402 2.936 0.262 3.755 0.631 10.15NG(1,1) 9.229 1.332 0.402 2.914 0.262 3.974 0.631 9.974G(1,1)-M 1.502 0.060 0.408 3.021 0.262 3.915 0.628 11.03EG(1,1) 7.064 0.988 14,013.14 4.402 0.407 5.665 0.826 21.30IG(1,1) 8.529 1.288 0.290 3.586 0.359 7.101 0.641

Table II(b). Estimation results of rate of returns for five stock market indices

Index ModelParameter estimates

d t-ratio y t-ratio

Composite G(1,1)-M 0.482 2.446EG(1,1) 0.060 0.530

Tins G(1,1)-M 0.489 1.951EG(1,1) 0.128 1.038

Plantations G(1,1)-M 0.402 2.605EG(1,1) 0.175 2...

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