forecasting stock market volatility using nonlinear) garch models

8/14/2019 Forecasting Stock Market Volatility Using Nonlinear) Garch Models

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Journa l of F orecasting. Vol. 15. 229-235 (199

Forecasting Stock Market Volatility Using(Non-Linear) Garch ModelsPHILIP HANS FRANSES A ND DICK VAN DIJKErasmus University, Rotterdam, The Netherlands

ABSTRACTIn this papeT we study the performance of the GARCH model and two its non-linear modifications to forecast weekly stock market volatility. Tmodels are the Quadratic GARCH (Engle and Ng. 1993) and the GlosteJagannathan and Runkle (1992) models which have been proposed describe, for example, the often observed negative skewness in stomarket indices. We find that the QGARCH model is best when testimation sample does not contain extreme observations such as the 19stock market crash and that the GJR model cannot be recommended fforecasting.KEY WORDS forecasting volatility; non-linear GARCH

INTRODUCTION

A stylized fact of financial time series is that aberrant observations seem to cluster in the senthat there are periods where volatility is larger than in other periods. Typically, these volatiperiods correspond to major (econom ic) events such as stock market crashes and oil criseAlthough most evidence in empirical finance indicates that retums on financial assets seeunforecastable at short horizons (see e.g. Granger. 1992, for a recent survey), the curreconsensus is that the variance of retums can be predicted using particular time series modelWithin this class of models, the Generalized Autoregressive Conditional Heteroscedastici(GARCH) model proposed by Engle (1982) and Bollerslev (1986) seems to be the mosuccessful (see Bollers lev, Chou and Kroner, 1992, for a survey of GARCH applicationsRoughly speaking, in a GARCH process the error variances can be modelled by aAutoregressive Moving Average ( A R S I A ) type process. A useful feature of the GARCH modis that it can effectively remove the excess kurtosis in retums.A further stylized fact is that the distribution of retums can be skewed. For example, fosome stock market indices, retums are skewed to the left. i.e. there are more negative thapositive outlying observations. The intrinsically symmetric GARCH model cannot cope wisuch skewness and, hence, one can expect that forecasts and forecast error variances fromGARCH model may be biased for skewed time series. Recently, a few modifications to thGARCH m odel have been proposed , which expbcitly take account of skewed distributions. our paper we consider two such modifications, the Quadratic GARCH model (QGARCHproposed by Engle and Ng (1993) (see also Sentana (1995) for a recent discussion of QGAR Cmodels), and the model advocated in Glosten, Jagannathan and Runkle. 1992, to be abbreviate

CCC 0277-6693/96/030229-07 Received January 199


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230 Journal of Foreca sting Vol. 15. Iss. No. 3as GJR ' The focus is the out-of-sample forecasting performance of these two non-lineamodels relative to the standard GARCH model, and on the performance of all three modelrelative to the simple random walk forecasting scheme. We limit our analysis to forecastingvolatility and not the mean of the time series.In the next section we present the models used in our modelling and forecasting exercise. Inthe third section we discuss the stock market data we use in our empirical study. In the fourthsection we discuss some within-sample estimation results. In the fifth section we evaluate thforecasting performance of the GARCH, QGARCH and GJR models as well as the RandomWalk model. In the final section we present conclusions.

GARCH MODELSConsider a stock market index /, and its return r,, which we construct as r,= log/, - log /,.,. Ithis paper the index / denotes weekly observations. Although higher-order models aresometimes found to be useful, the dominant empirical model for r, is the autoregression oforder p with GAR CH(1,1) disturbances iARip) - GA RC H(1,1)), which can be expressed as

e,~NiO.h,) (1h,= Q)-t-ael,-*-fih,.^

where B is the backward shift operator defined by B*jc, = x, .t. The parameter ^ reflects a constanterm, which in practice is typically estimated to be close or equal to zero. Furthermore, p iusually 0 or sm all, suggesting that there are usually no opportunities to forecast r, from its ownpast. It is assumed that the solutions of the characteristic equation 0 (z) = 0 lie outside the unicircle and that a), a. fi>0 and a + fi


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p . H. Franses and D. van D ijk Fore casting stock mark et volatilitywhere D,_^ is a dummy variable which takes a value of 1 when e ,. | < 0 and value of 0 whc,- , ^ 0 . Similar to the QGARCH m odel, when 5>Q, negative shocks will have a larger impon h, than positive shocks. Stationarity and stability of these models is discussed in the relevreferences. The QGARCH and GJR models can improve upon the standard symmetric GARCsince they can cope with negative (or positive) skew ness, the latter depending on the sign of additional parameter.

THE DATA AND RESEARCH M ETHODThe data we analyse in this paper are weekly observed indices for the stock markets in Germa(DAX), The Netherlands (EOE), Spain (MAD), Italy (MIL) and Sweden (VEC). The daspan 9 years, with the first observation being week 1 in 1986 and the last observation beiweek 52 in 1994. In our case, the weekly observations are taken as the values that are recordon W ednesdays.

A summary of some characteristics of the r, series is given in Table I. The number observations equals 469 for all five stock markets. The mean and variance are all quite smaThe excess kurtosis of the series exceeds 0, indicating the necessity of fat-tailed distributionsdescribe these variables. A key feature o f Table I, however, is that the estimated m easure skewness is large and negative. Especially for the DAX and EOE indices, skewness is large an absolute sense.The approach taken in this paper is as follows. Since our m ain goal is to evaluate the volatilforecasting performance of the three GARCH m odels, we wish to consider a reasonably larhold-out sample. We thus choose four years of observations to estimate the various modparameters. Furthermore, since it is not a priori assumed that one model necessarily dominaother models over the whole sample, we repeat our modelling and forecasting exercise fdifferent subsamples. We thus fit the models to a sample of four years , generate a one-steahead forecast, delete the first observation from the sample and add the next one, and generaagain a one-step-ahead forecast. In order to evaluate possibly changing patterns over the yeawe evaluate forecasting performance for the years 1990 to 1994. This results in an evaluation five times about 52 forecasts. Since we wish to minimize the impact of outlying observationsforecast evaluation, we use the Median of Squared Error (MedSE) instead of the usual MeSE.

Table I. Summary statistics of data on retums. The sample covers weekly observations for the years 19to 1994Stock marketGermany (DAX)Holland (EOE)Spain (MAD)Italy (MIL)Sweden (VEC)

n469469469469469

Mean(xlO-")9.2579.33522.2256.98123.848

Variance( x l O - ")7.8896.4519.3339.9379.836

Skewness- 0 . 9 7 2- 1 . 4 5 5- 0 . 5 3 1- 0 . 1 6 2- 0 . 3 4 8

Excekurto4.869.233.410.535.10

Source of data: Datastream.The weekly data concern Wednesday observations. The notation (10'*) means that the reported value should multiplied by 10"*


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232 Journal of Forecasting Vol. 15, Iss. No. 3WITHIN-SAMPLE ESTIMATION

In this section w e present som e w ithin-sample estimation results to giv e an idea of the possibleusefu lness o f (non-linear) GARCH m ode ls. To save space we consider the sample covering theperiod 1986 to 198 9, w hich giv es an indication of the typical estimated parameter valuesNotice that this period contains the stock market crash in 1987, which effects the skewness ofthe data.

In Table II we report the relevant parameter estimates for the GARCH model (1). Next to theparameter estimates, we report the value of the AIC and the value of the Log Likelihood (LnL)W e present these A IC and LnL values to compare m odels (1 ), (2) and (3). W e further report thevalues of Box-Pierce statistics for ijh]'^ and if/h,in order to check the empirical validity ofthe models. It is clear from Table II that the parameters a and /? in the GA R C^ (1,1 ) model areusually significant at the 5% level, and hence that the constant variance model can be rejected, aleast within sample. Furthermore, for two case s the sum of the a and fi parameters is close tounity. However, for no index do we observe that a + 0>\.In Table III we report som e estimation results for the non-linear variants o f the GARC H (1,1model. We report the estimates for the y parameter in the QGARCH model and the d parametein the GJR model. Although the results should be interpreted with care since not all non-normality may have been removed, the r-ratios in Table III indicate that the y and 6 parameterare significant at the 10% level at least for the D A X , EO E and VEC indices. The po ssibleusefu lness of non-linear modifications to the linear GAR CH mo del seem s to be confirmed bythe LnL values, although the AIC values do not suggest a clear favourite. Unreported estimatesof y and d in other samples shows that these parameters are not always relevant. Hence, therelevance o f these parameters can depend on one or two observations being included in ordeleted from the sample.

Table n. Estimation results forGARCHmodels for 1986-89

IndexDA XEO EM ADMILVEC

P00110

Parameter estima tes0)

0.444(3.55)0.185(2.75)0.011(1.43)0.038(1.40)0.273(2.32)

a0.267(5.17)0.283(4.65)0.128(3.40)0.129(2.71)0.222(2.71)

0.212(1.24)0.488(4.83)0.855(23.14)0.820(12.08)0.418(1.94)

AIC-7.00-7.01-6 .78-6.97-6.91

Diagnostics'

ln L447.93461.46442.99450.57456.98

(2.(10)13.07.692.597.234.11

(22(10)10.18.887.498.701.85

Notes:/-ratio are given in parentheses. Estimation is carried out using our own program written in Gau ss.The model is


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P H. Franses and D. van Dijk Forecasting stock market volatilityTable m . Estimation results of non-linear GARCH models for 1 9 86 -9

QGARCH GJR

IndexDAXEOEM ADMILVEC

Y(x lO- ' )0.022(1.82)0.115(1.73)-0.125(-1.57)-0.012(-0.23)0.199(2.19)

A IC-7.00-7.01-6.78-6.97-6.91

LnL'450.06463.86444.12450.61458.07

d0.482(4.52)0.202(1.85)-0.104(-1.05)-0.023(-0.32)0.274(2.02)

AIC-7.00-7.01-6 .78-6 .97-6 .91

L4546444545

Note:The QGARCH model is given by (B)r, = fi + E,,e,-NiO,h,),h, = (o +a(,.,-yy +^h,., and the GJR mo^(B)r, = ft + e,, e, ~ N(0, h,), h, = (t) + aej.f + dD,,t - ej.t + fih,.,, where >,. , = 1 when c, . ,< = O an d D ,. , 0 e, ., 0 , and where 0( f i) is an autoregressive polynomial of order p and r, is the weekly observed retum series./-ratios are given in parentheses. Estimation is carried out using our own program written in G auss.See Table n.

OUT-OF-SAMPLE FORECASTINGTo evaluate the ability of (non-linear variants of) GARCH models to adequately forecastvolatility in financial time series, we need a measure of the 'true vola tility' As is standarrelated studies on forecasting volatility (see Day and Lewis, 1992; Pagan and Schwert, 19we use the m easure w,, defined by

y^',= ir,-rywhere f is the average retum over the last four years of weekly observations. As a measureh hd fhe one-step-ahead forecast error ,^i we use

/ + l = H ' , ^ l - ^ , ^ , where ^,+, is generated using the h, equations in (1), (2) and (3) with the estimates forparameters a>, a, fi, y and 6 substituted. To compare the forecasts from the GARCH mowith a naive no-change forecast, we also evaluate the forecast errors for the Random W(RW ) mo del, for which forecasts are given by ^ ,^, = w,. Notice that we calculate one-step-ahforecasts, i.e. we use the estimated models to forecast for / + 1 while the models are calibrfor four years of observations until and including t.In Table IV we report the MedSE for the years 1990 to 1994. The results indicate thatQGARCH model is preferred in 11 of the 25 cases, and that the RW, GARCH and GJR moare preferred in 12, 2 and 0 cases, respectively. The results for the GJR model suggest that not a useful tool for forecasting. Overall, it seems that RW and QGARCH perform abequally well.The forecasts for 1990 and 1991 are based on models which are fitted from estimasamples which include the 1987 stock market crash observation. From Table IV we noticefor these two years, the RW m odel outperforms the other models in 8 of the 10 cases. Onother hand, when we consider the years 1992, 1993 and 1994. we find that the (JGARCH m


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23 4 Journal of Forecasting Vol. 15, Iss. No. Table IV. Out-of-sample forecasting performance of ARCH, QGARCH, GJR and random walk modefor the volatility o f stock market ind ices

IndexDAX

EO E

M AD

MIL

VEC

ModelGARCHQGARCHGJRRWGARCHQGARCHGJRRWGARCHQGARCHGJRRWGARCHQGARCHGJRRWGARCHQGARCHGJRRW

19904.3953.146178.4173.5661.3051.497126.6561.1463.2493.298

116.1161.1121.1301.08321.9231.1733.0273.46248.5202.483

19913.3593.32213.8210.5851.0501.07661.7130.1481.1181.085

125.2611.0622.3581.94310.1701.6642.3962.978218.7691.104

Forecast period19921.5201.4255.5850.2300.8480.7073.7790.0902.7171.839

23.4420.8593.3512.34965.2216.7363.0402.27967.6102.554

19931.7750.8776.7171.0130.7670.5522.5450.6332.0761.008

16.1931.2148.8956.04771.36419.8516.0675.3573.4971.621

19942.852.7875.493.210.921.0011.720.982.421.96

47.615.028.448.4735.1910.803.093.204.473.10

Note:As a measure of 'true volatility' we use (r, - r)^, where f is the averagereturnover r - 1 ,. .. , /-four years.The m odels are estimated for samp les of four years, where in each new sample the first observation is deleted and thnext observation is included, and one-step ahead forecasts are generated for 1990,..., 1994.Each cell contains the Median of Squared Errors (x 1 0'^) .outperforms its rivals in 9 of the 15 cases. Hence, the forecasting performance of the GARCHtype models appear sensitive to extreme within-sample observations.

CONCLUSIONSOur forecasting results for five weekly stock market indices show that the (JGARCH model casignificantly improve on the linear GARCH model and the no-change forecasting model, icases when the forecasting models are calibrated on data which exclude such extreme events athe 1987 stock market crash. The GJR model, on the other hand, cannot be recomm ended.

ACKNOWLEDGEMENTSA previous version of this paper was presented at the INFORMS meeting in Singapore (Jul1995). We thank Olaf van ThuU for his help with the data and computing eflForts. The first authothanks the Royal Netherlands Academy of Arts and Sciences for its financial support. We alsgratefully acknowledge the help of Tim Bollerslev and Hans Ole Mikkelsen with the compute


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P H. Franses andD. van Dijk Fore casting stock ma rket volatilityREFERENCES

Bollerslev, T., 'Generalized autoregressive conditional heteroskedasticity' Journal of Econometric(1986), 307-27.Bollerslev, T., Chou, R. and Kroner, K., ARCH modeling in finance: a review of the theory empirical eviden ce'. Journal of Econometrics. 52 (1992), 5-59.Day, T. and Lewis, C., 'Stock market volatility and the information content of stock index op tiJournal of Econometrics. 52 (1992), 267 -87 .Engle, R. F., 'Autoregressive conditional heteroskedasticity with estimates of the variance of inflation', Econometrica. 50 (1982), 987-1 008 .Engle, R. F. and Ng , V., 'Measuring and testing the impact of news on volatility '. Journal of Financ(1993), 1749-78.Glosten, L., Jagannathan, R. and Runkle, D., 'On the relation between the expected value andvolatility of nominal excess return on stock s'. Journal of Finance, 46 (1992), 1779-80 1.Granger, C. W. J., 'Forecasting stock market prices: Lessons for forecasters' International JournForecasting. 8 (1992), 3- 13 .Nelson, D., 'Conditional heteroskedasticity in asset retums: a new approach' Econometrica. 59 (19347-70.Pagan, A. R. and Schwert, G. W., Alternative models for conditional stock market vola tility'. JournaEconometrics. 45 (1990), 267 -90 .Sentana, E., 'Quadratic ARCH m odels'. Review of Economic Studies. 62 (1995), 63 9-61 .

Authors' biographies:Philip Hans Franses is a research fellow of the Royal Netherlands Academy of Arts and Scienaffiliated to the Econometric Institute, Erasmus University Rotterdam. His research in terests are mselection and forecasting in time series. He has published on these topics in, e.g., the JournaForecasting. Journal of Marketing Research. Journal of Econometrics, and the Journal of BusineEconomic Statistics.Dick van Dijk is a PhD student at the Tinbergen Institute, Erasmus University Rotterdam. His resetopic is model selection in non-linear time series.Authors' address:Philip Hans Franses (Econometric Institute) and Dick van Dijk (Tinbergen Institute), ErasUniversity, Rotterdam, PO Box 1738, N L- 30 00 DR Rotterdam, The Netherlands.


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forecasting stock market volatility using nonlinear) garch models

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