Forecasting stock market volatility using non linear garch models

Vol Forecasting Stock Market Volatility Using Non-Linear Garch Models PHILIP HANS FRANSES AND DICK VAN DIJK Erasmus University, Rotterdam, The Netherlands ABSTRACT In this papet we study the performance of the model and two of its non-linear modifications to forecast weekly stock market volatility.

The models are the Quadratic Engle and Ng. Jagannathan and Runkle models which have been proposed to describe, for example, the often observed negative skewness in stock market indices. We find that the model is best when the estimation sample does not contain extreme observations such as the stock market crash and that the model cannot be recommended for forecasting.

Typically, these volatile periods correspond to major economic events such as stock market crashes and oil crises. Although most evidence in empirical finance indicates that retums on financial assets seem unforecastable at short horizons see e. Within this class of models, the Generalized Autoregressive Conditional Heteroscedasticity model proposed by Engle and Bollerslev seems to be the most successful see Bollerslev, Chou and Kroner,for a survey of applications.

Roughly speaking, in a process the error variances can be modelled by an Autoregressive Moving Average ARSIA type process. A useful feature of the model is 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, for some stock market indices, retums are skewed to the left. The intrinsically symmetric model cannot cope with such skewness and, hence, one can expect that forecasts and forecast error variances from a model may be biased for skewed time series.

Recently, a few modifications to the model have been proposed, which expbcitly take account of skewed distributions. In our paper we consider two such modifications, the Quadratic model proposed by Engle and Ng see also Sentana for a recent discussion of modelsand the model advocated in Glosten, Jagannathan and Runkle. We limit our analysis to forecasting volatility and not the mean of the time series.

In the next section we present the models used in our modelling and forecasting exercise. In the third section we discuss the stock market data we use in our empirical study.

In the fourth section we discuss some within-sample estimation results. In the fifth section we evaluate the forecasting performance of the, and models as well as the Random Walk model.

In the final section we present conclusions. Furthermore, p is usually 0 or small, suggesting that there are usually no opportunities to forecast r, from its own past. AU odd moments of e, in model 1 equal zero, and hence e, and r, are symmetric time series with fat tails. For stock market time series, however, r, may display significant negative skewness. This empirical stylized result seems to be due to the fact that stock market crashes occur more frequently and more quickly than stock market booms, and that the absolute size of crashes is much larger.

Two simple classes of models that can cope with skewed returns are the Quadratic model proposed by Engle and Ng and the so-called model, advocated by Glosten, Jagannathan and Runkle Although we have considered this model as a possibly useful candidate for our purposes, it was found not to be veiy useful for repeated forecasting exercises. Given the latter purpose of repeatedly specifymg and estimating models, an obvious requirement of these models is that the estimation method is reasonably simple and that parameter convergence occurs reasonably quick.

Unfortunately, it has been our experience while ninning the estimation procedures that parameter estimation of the E model can be tedious In fact, m almost all cases, no quick convergence was obtained.

Only when we were able to generate a particular set of starting-values with a precision of 8 digits could we obtain convergence. Stationarity and stability of these models is discussed in the relevant references. The and models can improve upon the standard symmetric since they can cope with negative or positive skewness, the latter depending on the sign of the additional parameter.

THE DATA AND RESEARCH METHOD The data we analyse in this paper are weekly observed indices for the stock markets in Germany DAXThe Netherlands EOESpain MADItaly MIL and Sweden VEC.

The data span 9 years, with the first observation being week 1 in and the last observation being week 52 in In our case, the weekly observations are taken as the values that are recorded on Wednesdays. A summary of some characteristics of the r, series is given in Table I.

The number of observations equals for all five stock markets. The mean and variance are all quite small.

The excess kurtosis of the series exceeds 0, indicating the necessity of fat-tailed distributions to describe these variables. A key feature of Table I, however, is that the estimated measure of skewness is large and negative. Especially for the DAX and EOE indices, skewness is large in an absolute sense. The approach taken in this paper is as follows.

Since our main goal is to evaluate the volatility forecasting performance of the three models, we wish to consider a reasonably large hold-out sample. We thus choose four years of observations to estimate the various model parameters. Furthermore, since it is not a priori assumed that one model necessarily dominates other models over the whole sample, we repeat our modelling and forecasting exercise for different subsamples.

We thus fit the models to a sample of four years, generate a one-stepahead forecast, delete the first observation from the sample and add the next one, and generate again a one-step-ahead forecast. In order to evaluate possibly changing patterns over the years, we evaluate forecasting performance for the years to This results in an evaluation of five times about 52 forecasts.

Since we wish to minimize the impact of outlying observations on forecast evaluation, we use the Median of Squared Error MedSE instead of the usual Mean SE. Summary statistics of data on retums. The sample covers weekly observations for the years to Stock market n Mean xlo-" Variance xlo-" Skewness Excess kurtosis Germany DAX Holland EOE Spain MAD Italy MIL Sweden VEC Source of data: The weekly data concern Wednesday observations.

To save space we consider the sample covering the period towhich gives an indication of the typical estimated parameter values. Notice that this period contains the stock market crash inwhich effects the skewness of the data.

In Table II we report the relevant parameter estimates for the model 1. Next to the parameter estimates, we report the value of the AIC and the value of the Log Likelihood LnL. We present these AIC and LnL values to compare models 12 and 3.

Furthermore, for two cases the sum of the a and fi parameters is close to unity. In Table III we report some estimation results for the non-linear variants of the 1,1 model. We report the estimates for the y parameter in the model and the d parameter in the model. The possible usefulness of non-linear modifications to the linear model seems to be confirmed by the LnL values, although the AIC values do not suggest a clear favourite.

Unreported estimates of y and d in other samples shows that these parameters are not always relevant. Hence, the relevance of these parameters can depend on one or two observations being included in or deleted from the sample. Estimation results for models for Parameter estimates Diagnostics' Index P 0 a AIC lnl 2.

Estimation is carried out using our own program written in Gauss. The diagnostics are the Akaike Information Criterion [AIC], which will be used to compare these models with those in Table III. The AIC is calculated so that the model with the smallest AIC value is preferred. Estimation results of non-linear models for Index Y xlo-' AIC LnL' d AIC LnL DAX EOE MAD MIL VEC 1.

Notice that we calculate one-step-ahead forecasts, i. In Table IV we report the MedSE for the years to The results indicate that the model is preferred in 11 of the 25 cases, and that the, and models are preferred in 12, 2 and 0 cases, respectively.

The results for the model suggest that it is not a useful tool for forecasting. Overall, it seems that and perform about equally well. The forecasts for and are based on models which are fitted from estimation samples which include the stock market crash observation. From Table IV we notice that for these two years, the model outperforms the other models in 8 of the 10 cases. On the other hand, when we consider the yearsand we find that the J model.

Out-of-sample forecasting performance of ARCH, and random walk models for the volatility of stock market indices Forecast period Index Model DAX EOE MAD MIL VEC Note: The models are estimated for samples of four years, where in each new sample the first observation is deleted and the next observation is included, and one-step ahead forecasts are generated forHence, the forecasting performance of the type models appear sensitive to extreme within-sample observations.

The model, on the other hand, cannot be recommended. We thank Olaf van ThuU for his help with the data and computing eflforts. The first author thanks the Royal Netherlands Academy of Arts and Sciences for its financial support. We also gratefully acknowledge the help of Tim Bollerslev and Hans Ole Mikkelsen with the computer programs. Finally, we thank the Editor and an anonymous referee for useful suggestions.

Journal of Finance, 46Granger, C. Lessons for forecasters' International Journal of Forecasting. Review of Economic Studies.

Philip Hans Franses is a research fellow of the Royal Netherlands Academy of Arts and Sciences, affiliated to the Econometric Institute, Erasmus University Rotterdam. His research interests are model selection and forecasting in time series.

He has published on these topics in, e. Journal of Marketing Research. Journal of Econometrics, and the Journal of Business arid Economic Statistics. Dick van Dijk is a PhD student at the Tinbergen Institute, Erasmus University Rotterdam.

His research topic is model selection in non-linear time series. Philip Hans Franses Econometric Institute and Dick van Dijk Tinbergen InstituteErasmus University, Binary option analysis, PO BoxNL DR Rotterdam, The Netherlands.

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