Forecasting Volatility in the Chinese Stock Market under Model Uncertainty 1

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1 Forecasting Volatility in the Chinese Stock Market under Model Uncertainty 1 Yong Li 1, Wei-Ping Huang, Jie Zhang 3 (1,. Sun Yat-Sen University Business, Sun Yat-Sen University, Guangzhou, 51075,China) (3.Institute of China s Economic Reform Development, Renmin University of China) Abstract Volatility forecasting is an important issue in empirical finance. he main purpose of this paper is to apply the model averaging technique to reduce volatility model uncertainty for improving volatility forecasting. Six GARCH-typed models are considered as candidate models for model averaging. Based on some statistical loss functions, the results show that the combined forecasts can be a better approach than the individual forecasts in the Chinese stock market which is the largest emerging market in the world. Keywords: China s financial market; volatility forecasting; forecast combination; forecast accuracy 1. Introduction Accurate volatility forecasting is one of the key tasks in empirical finance, such as, in the applications of investment, the security valuation, risk management, and monetary policy. Consequently, in the past two decades, forecasting volatility in financial markets has attracted growing attention by academics and practitioners. here are many models which can be used for forecasting asset volatilities. Engle(198) firstly proposed the Autoregressive Conditional Heteroscedasticity (ARCH) model. A generalization of ARCH named GARCH was developed by Bollerslev(1986). Now, many general extensions of these original models have been developed, see Francq and Zakoian (010) for a collection of studies for all kinds of ARCH-types volatility models. An excellent review about volatility forecasting using these volatility models is recently given by Poon and Granger (003). While the use of models has undeniably led to a better measurement of volatility, it has in turn given rise to a new problem, known as model risk or model uncertainty, linked to the uncertainty on the choice of the volatility model itself. In the literature, it is known that discarding model uncertainty can face a large utility or wealth loss, see Avramov(00) and Rapach, Strauss and Zhou (009) reference therein. However, with ignoring model uncertainty, most of empirical studies on 1 Corresponding authors: Jie. Zhang, Institute of China s Economic Reform Development, Renmin University of China, Beijing 10087, China; zhangjie040@ruc.edu.cn

2 volatility forecasting focus on choosing a best model among the candidate models using some techniques ranging from in-sample criterions through out-of criterions. In this paper, instead of choosing a best model, we use the model averaging technique to deal with model uncertainty. Several volatility models are considered as candidate models for model averaging. In addition, consider that there are a lot of empirical studies on stock market volatility internationally, but little on the emerging stock markets, we apply the model averaging technique into the China s stock market. o the best of our knowledge, this is the first study to explore model averaging technique to forecasting the China s stock market volatility under model uncertainty. his study attempts to enrich the existing literature by investigating the case of China, the largest transitional economy in the world, which has a very unique market structure including the dominance of individual investors over institutional investors on the stock market (Ng and Wu, 007) he rest of this paper is organized as follows. Section describes the data and model averaging approach. Section 3 is the empirical results and forecasting valuation. Finally, some conclusions and discussions are included in section 4.. Data and methodology.1 data China has two stock exchanges, Shanghai stock exchange and Shenzhen stock exchange, which were respectively established on December 19, 1990 and July 3, Large companies mainly go public at Shanghai stock exchange while middle and small companies at Shenzhen stock exchange. As one of the biggest emerging markets in the world, the Chinese stock market,including Shanghai and Shenzhen stock exchange, comprises 063 listed companies by the end of 010, which have about 3050 billion yuan market value. he China Securities Regulatory Commission (SCRC) chooses and approves the companies for listing on the exchange. he exchange regulates the trading and has no authority in the selection of companies for listing. Moreover, investors must maintain a capital account with a securities dealer and can only trade up to the limit of their capital. Margin trading and short sale are prohibited. he stock market trading system experienced great changes. During the period from May 1, 199 to December 15, 1996 its price-ceiling was abolished and stock price was determined by the force of demand and supply. 3 On December 16, 1996, Shanghai and Shenzhen stock exchange put the 10% limit-up and limit-down pricing system into practice. he change of trading system can make the model structure brake. If the data of period from 199 to 1996 are included, the volatility will be overestimated. Hence, in this By the end of 010, the New York stock exchange has 317 listed companies and billion dollar market value, which converted a total of trillion yuan for 6.67yuan per dollar. 3 On May 1, 199 the Shanghai stock exchange abolished the limit-up and limit-down price system so that the Shanghai stock market index rose more than 104% that day, the largest increase experienced.

3 paper, we only select the Shanghai A index which cover all A-share listed stocks after 1996, weighted by the market capitalization without dividends reinvested and takes December 19, 1990 as the base of 100. he raw data are daily stock price index ( p t ) covering the period from January, 1997 to December 31, 010, making a total of 3385 daily observations. Daily returns are identified as the first difference in the natural logarithm of the closing index value for two consecutive trading days, i.e, rt ln( pt / pt 1). According to Merton (1980) and Perry (198), the realized volatility in a month can be simply calculated as the sum of squared daily returns in corresponding month, N t rt t1 (1) able 1 contains some summary statistics. We can find that the skewness is negative, which indicates the distribution is non-symmetric. Moreover, the large kurtosis suggests that the return series are leptoturtic (fat tailed) and sharply peaked about the mean compared with the normal distribution. Also the JB statistic rejects the null hypothesis of normal distribution. able 1: Sample statistics for daily returns (January, December 31, 010) observations mean(%) median(%) max(%) min(%) sd skewness kurtosis JB *** Note: *** denotes significance at the 1% level.. model averaging methodology he standard GARCH(1,1) model is often used to forecast the asset volatility. We model the rt as rt t, assuming the conditionally normally distribution forecast errors( t ) with zero mean and variance t. he GARCH (1, 1) evaluate positive and negative t of the same magnitude equally which is given by () t 0 1 t1 t1 he GARCH model cannot explain asymmetry in distribution of stock returns. he EGARCH(1,1) and GJR-GARCH(1,1) models allow asymmetry in the conditional volatility equation as follows: t 1 t1 ln( t ) 0 ln( t1) t1 t1 D (4) t 0 1 t1 t1 t1 t1 (3) where D t 1is a dummy variable taking the value of 1 if 1 0, and 0 otherwise. In empirical finance, the return distribution is often shown as fatter tails, see the t

4 summary statistics in able 1. he conditional normality in GARCH models can be replaced by t distributions. Hence, in this paper, we introduce six models for model averaging, that is, GARCH(1,1), GARCH(1,1)-t, EGARCH(1,1), EGARCH(1,1)-t, GJR-GARCH(1,1) and GJR-GARCH(1,1)-t. he data is divided into two parts, one is used for training set to estimate the parameters, the other is used to evaluate the forecasting effect. In order to evaluate the forecast effect, we consider three different choice of a training sample. In the first case, the first 84 month observations (from January 1997 to December 003) are used for training samples; in the second case, the first two-thirds (from January 1997 to April 005), 11 monthly observations, are used; in the third cases, the first three quarters (from January 1997 to June 006), 16 monthly observations, are used. We make a one-step-ahead forecast of the day s volatility, and then roll the sample forward one observation at a time, constructing a new one-step-ahead forecast at each stage. hese out-of-sample forecasts of daily variance are summed up to obtain monthly total volatility. In this study, we use model averaging strategies to generate the forecasts. he combination forecasts of ^ c, individual forecasts given as follows: made at month are weighted averages of the N ˆ N ˆ c, i, i, i1 (5) where {, } N i i 1are the weights formed at month. ^ i, are the single forecasts of each individual model at month. Five different averaging strategies are adopted to generate the forecasts as follows: 1) the simple mean weight averaging: 1 i, for i1,, N N ) the median averaging: the median of { ˆ } N i, i 1 is used 3) the trimmed mean averaging: set i, 0 for the individual forecasts with the smallest and largest magnitudes and, i 1 ( N ) for the remaining individual forecasts 4) the regression combination approach by Clements and Hendry(1998): the combination weights are derived from the following regression: ˆ ˆ ˆ (6) 0 1 1,, N N, where is the monthly realized volatility in equation (1) and ˆi,, i1,,, Nare N different forecasts of individual forecasts. he resulting combination forecast is then given by ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (7) c, 0 1 1,, N N,

5 where ˆc, is the combination forecast. We consider the OLS fixed weights. he fixed weights allows for the initial period of 0 monthly forecast to be used for estimating weights while the remaining forecasts are used for the purpose of comparison. 5) the OLS time varying weights are also considered using regression-based averaging approach. For the OLS time varying weights, to form a combination forecast at month, combination weights are obtained by estimating Eq(6) on forecasts from 0 to 1,combining the various individual forecasts ˆi, formed at month using these weights and then rolling the window of forecasts gets a new combinations weights. 3. Empirical result In this paper, we evaluate the forecast performance by the symmetric and asymmetric statistical loss functions. he symmetric loss functions are the mean absolute error (MAE), the root mean squared error (RMSE) and the mean absolute percentage error (MAPE) given by: 1 MAE I ˆ (8) 1 1 RMSE ( ( ) ) I I ˆ 0.5 (9) 1 ˆ MAPE I (10) 1 1 he asymmetric loss functions can be expressed as follows: O U 1 MME( U) ˆ ˆ 1 1 (11) O U 1 MME( O) ˆ ˆ 1 1 (1) where OU ( ) is the number of over-(under)-predictions. MME( O) and MME( U) penalize the over-predictions and under-predictions more heavily, respectively. he main results are presented in tables,3 and 4. he results show that no individual forecast consistently outperforms all other forecast across all statistic loss functions for all forecast cases. It is demonstrated that we cannot always choose a best model to forecast the stock market volatility. If only one model is used for forecast, it has in turn given rise to a new problem, known as model risk or model

6 uncertainty. In the first forecast case (see table ), according to the statistic loss functions, the OLS fixed weights combination almost outperforms its competitors and its performance is at least 45% better compared to the worst model. he OLS time varying weights combination rank second. Forecast combination tends to be more useful for forecasting volatility. If the under-predictions are more heavily penalized, the GARCH(1,1) and GJR-GARCH(1,1) models rank first and second respectively. In addition, the EGARCH(1,1)_t and GJR-GARCH(1,1) models are the worst performing models. Forecast combination can therefore reduce the risk of complete forecasting failure. Moreover, it can be found that the combined forecasts are better than the average loss. able : symmetric and asymmetric loss function for individual and combination forecasts (forecast period from January 004 to December 010, the latter 64 monthly forecasts are used for accuracy comparison) 100 MAE 100 RMSE MAPE 10 MME(U) 10 MME(O) Models Value Rank Value Rank Value Rank Value Rank Value Rank U(%) O(%) GARCH(1,1) % 65.60% GARCH(1,1)_t % 60.90% EGARCH(1,1) % 53.10% EGARCH(1,1)_t % 50.00% GJR-GARCH(1,1) % 65.60% GJR-GARCH(1,1)_t % 59.40% Average loss Mean % 59.40% Median % 59.40% trimmed mean % 59.40% OLS fixed weights % 3.80% OLS time varying weights % 48.40% Note: U(%) and O(%) give the percentage of under-predictions and over-predictions. he best performing model has a rank 1. As mentioned above, the first 84 monthly volatilities are used for estimation, while the latter 84 ones are used for forecasts (the first 0 forecasts are used to estimate the weights, whereas the rest are used for accuracy comparison). he average loss is the mean loss of MAE, RMSE, MAPE, MME(U) and MME(O) for the six individual models. Mean, median, trimmed mean, OLS fixed weights and OLS time varying weights are five model combination methods. able 3: symmetric and asymmetric loss function for individual and combination forecasts (forecast period from May 005 to December 010, the latter 36 monthly forecasts are used for accuracy comparison) 100MAE 100RMSE MAPE 10MME(U) 10MME(O) Models Value Rank Value Rank Value Rank Models Value Rank Value Rank Value GARCH(1,1) % 5.80% GARCH(1,1)_t % 47.0% EGARCH(1,1) % 47.0% EGARCH(1,1)_t % 47.0%

7 GJR-GARCH(1,1) % 55.60% GJR-GARCH(1,1)_t % 55.60% Average loss Mean % 5.80% Median % 5.80% trimmed mean % 5.80% OLS fixed weights % 50.00% OLS time varying weights % 58.30% Note: U(%) and O(%) give the percentage of under-predictions and over-predictions. he best performing model has a rank 1. As mentioned above, the first 11 monthly volatilities are used for estimation, while the latter 56 ones are used for forecasts (the first 0 forecasts are used to estimate the weights, whereas the rest are used for accuracy comparison). he average loss is the mean loss of MAE, RMSE, MAPE, MME(U) and MME(O) for the six individual models. Mean, median, trimmed mean, OLS fixed weights and OLS time varying weights are five model combination methods. able 4: symmetric and asymmetric loss function for individual and combination forecasts (forecast period from July 006 to December 010, the latter monthly forecasts are used for accuracy comparison) 100MAE 100RMSE MAPE 10MME(U) 10MME(O) Models Value Rank Value Rank Value Rank Model Value Rank Value Rank Value GARCH(1,1) % 7.70% GARCH(1,1)_t % 7.70% EGARCH(1,1) % 63.60% EGARCH(1,1)_t % 63.60% GJR-GARCH(1,1) % 7.70% GJR-GARCH(1,1)_t % 68.0% Average loss Mean % 68.0% Median % 68.0% trimmed mean % 68.0% OLS fixed weights % 86.40% OLS time varying weights % 63.60% Note: U(%) and O(%) give the percentage of under-predictions and over-predictions. he best performing model has a rank 1. As mentioned above, the first 16 monthly volatilities are used for estimation, while the latter 4 ones are used for forecasts (the first 0 forecasts are used to estimate the weights, whereas the rest are used for accuracy comparison). he average loss is the mean loss of MAE, RMSE, MAPE, MME(U) and MME(O) for the six individual models. Mean, median, trimmed mean, OLS fixed weights and OLS time varying weights are five model combination methods.

8 In the second forecast case (see table 3), four statistic loss functions favor the OLS time varying weights combination. In the third forecast case (see table 4), the OLS time varying weights combination almost outperforms its competitors according to the statistic loss function with the exception of MME(U). he best combination forecasts performances are at least 0% better than the worst single model in both cases. From the examination of above results it indicates that the combination forecasts deliver a statistically advantage. Although combination forecasts do not always beat the best single-model forecasts, our empirical results show that almost all the statistic loss functions of combined forecasts are smaller than those of the worst single-model forecasts and the average single-model forecasts for all cases. It not surprising, as different models can obtain different dynamics in volatility. 4. Conclusion his paper examines six individual models and five combined strategies for forecasting the China s stock market volatility. We find that no individual forecast consistently outperforms all other forecast across all statistic loss functions under all the considered cases. In addition, the best model among of candidate models chosen for forecasting is seriously dependent on the choice of treating data. If the combined methods are used, the results show that the combined forecasts are more accurate than the worst single-model forecasts for all cases. At last, we do not consider other exogenous variables, such as inflation rate, lagged market trading volume. hese questions are beyond the discussion in our paper and may be regarded as future research orientation. Acknowledgment he research is supported by Chinese National Science Fund under No

9 Reference Bollerslev,.(1986). Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, Clements, M.P. and D.F. Hendry (1998). Forecasting Economic Processes. International Journal of Forecasting, 14, Doron Avramov, D. (00). Stock return predictability and model uncertainty, Journal of Financial Economics,.64(3), 00, Engle, R. F.(198). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, Francq,C. and Zakoian,J.M. (010). GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley publishing house. Merton, R.(1980). On estimating the expected return on the market: An explanatory investigation. Journal of Financial Economics, 8, Ng, L., Wu, F.(007). he trading behavior of institutions and individuals in Chinese equity markets, Journal of Banking & Finance, 31, Perry, P.(198).he time-variance relationship of security returns: Implications for the return-generating stochastic process, Journal of Finance, 37, Poon, S.H., and Granger, C.W.J. (003). Forecasting the volatility in financial market: a review, Journal of Economic Literature, 41, Rapach,D.E.,Strauss,J.K. and Zhou,G.F.(009). Out-of-Sample Equity Premium Prediction: Combination Forecasts and Links to the Real Economy. Review of Financial Study, 3 (),

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