# Modeling and Forecasting Volatility of the Malaysian Stock Markets

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Modeling and Forecasting Volatility of the Malaysian Stock Markets

Journal of Mathematics and Statistics 5 (3):234-240, 2009 ISSN 1549-3644 © 2009 Science Publications Modeling and Forecasting Volatility of the Malaysian Stock Markets Ahmed Shamiri and Zaidi Isa School of Mathematical Sciences, University Kebangsaan Malaysia, 43600 UKM, Bangi, Malaysia Abstract: Problem statement: One of the main purposes of modeling variance is forecasting, which is crucial in many areas of finance. Despite the burgeoning interest in and evaluation of volatility forecasts, a clear consensus on witch volatility model/or distribution specification to use has not yet been reached. Therefore, the out of-sample forecasting ability should be a natural model selection criterion for volatility models. Approach: In this study, we used high-frequency to facilitate meaningful comparison of volatility forecast models. We compared the performance of symmetric GARCH, asymmetric EGARCH and non leaner asymmetric NAGARCH models with six error distributions (normal, skew normal, student-t, skew student-t, generalized error distribution and normal inverse Gaussian). Results: The results suggested that allowing for a heavy-tailed error distribution leads to significant improvements in variance forecasts compared to using normal distribution. It was also found that allowing for skewness in the higher moments of the distribution did not further improve forecasts. Conclusion: Successful volatility model forecast depended much more heavily on the choice of error distribution than the choice of GARCH models. Key words: GARCH-models, asymmetry, stock market indices and volatility modeling INTRODUCTION Traditional regression tools have shown their limitation in the modeling of high-frequency (weekly, daily or intra-daily) data. Assuming that only the mean response could be changing with covariates while the variance remains constant over time often revealed to be an unrealistic assumption in practice. This fact is particularly obvious in series of financial data where clusters of volatility can be detected visually. During the last few decades we have seen a multitude of different suggestions for how to model the second moment, often referred to as volatility, of financial returns. Indeed, it is now widely accepted that high frequency financial returns are heteroskedastic. Among the models that have proven the most successful are the Auto-Regressive Conditional Heteroskedasticity (ARCH) family of models introduced by Engle[4] and the models of Stochastic Variance (SV) pioneered by Taylor[14]. One of the main purposes of modeling variance is forecasting, which is crucial in many areas of finance such as option pricing, value at risk applications and portfolio selection. Therefore, the out of-sample forecasting ability should be a natural model selection criterion for volatility models. The vast majority of variance forecasting articles have used squared daily returns as the proxy for ex post variance. This is, as shown by Andersen and Bollerslev[2], an unbiased but exceedingly noisy estimator. While the literature that examines competing variance models is abundant, very little work has been done comparing different distribution assumptions, with the noticeable exceptions of Anders[1] and Shamiri[10,11]. However, none of these papers has explicitly focused on evaluating asymmetric GARCH models forecast using different error distributions. The previous studies paid more attention on the symmetric GARCH model, while in this study we focus on both symmetric and asymmetric volatility models as well as symmetric and asymmetric distributions. Unfortunately, ARCH models often do not fully capture the thick tails property of high frequency financial time series. However, another striking characteristic of high-frequency financial returns is that they are often characterized by fat-tailed distribution. In fact, the kurtosis of most asset returns is higher than three, which means that extreme values are observed more frequently that for the normal distribution. While the high kurtosis of the returns is a well-established fact, the situation is much more obscure with regard to the symmetry of the distribution. Corresponding Author: Ahmed Shamiri, School of Mathematical Sciences, University Kebangsaan Malaysia, 43600 UKM, Bangi, Malaysia 234 J. Math. & Stat., 5 (3):234-240, 2009 EGARCH: Nelson[8] introduced the first asymmetric GARCH model known as exponential GARCH model (EGARCH). This model looks at the conditional variance and tries to accommodate for the asymmetric relation between stock returns and volatility changes. Nelson implements that by including an adjusting function g (z) in the conditional variance equation, it in turn becomes expressed by: Many authors do not observe anything special on this point, but other researchers, such as Simkowitz and Beedles[13] and Kon[7] have drawn the attention to the heavy tail of the distribution. Shamiri[10,12] have shown that a fat-tailed distribution is required for modeling daily returns of East Asian equity markets. This study adds to the literature in several important directions. Firstly, in this study we demonstrate that this gap can be filled by a rigorous density forecast models comparison methodology. Secondly, we compare the performance of the GARCH, EGARCH and NAGARCH models and also introduce different densities (Normal, Skew normal, Student-t, Skew student-t, Generalized Error Distribution GED and Normal Inverse Gaussian (NIG)). Thirdly, using high-frequency data from Kuala Lumpur Composite Index (KLCI) of Malaysian’s stock market to facilitate meaningful comparison of the forecast results. In the model estimation, it is shown that allowing for a leptokurtic return distribution significantly improves the fit of the model. In terms of out-of-sample forecasting performance, allowing for excess kurtosis leads to significant improvements over the normal distribution, whereas allowing for non-centrality does not further enhance forecasts. g(z t ) = (2) 2 θ1z t N (3) + θ2 [ z t − E z t ] magnitude effect Notice moreover that E|zt| depends on the assumption made on the unconditional density. The EGARCH model differs from the standard GARCH model in two main aspects. First, it allows positive and negative shocks to have a different impact on volatility. Second, the EGARCH model allows large shocks to have a greater impact on volatility than the standard GARCH model. NAGARCH: Engle[5] proposed the Asymmetric NonLinear GARCH Model (NAGARCH), given by: p ( σ2t = α 0 + ∑ α i ε t −1 + λσ 2t −1 i =1 q ) + ∑β σ 2 j =1 j 2 t −1 (4) Failure to capture fat-tails property of highfrequency financial time series has led to the use of non-normal distributions to better model excessive third and fourth moments. Since it may be expected that excess kurtosis and skewness displayed by the residuals of conditional heteroskedasticity models will be reduced when a more appropriate distribution is used, we consider six distributions in this study: the normal, skew-normal, student-t, skew-student-t, GED and NIG. In the following, f (z) is the standardized density function of the standardized residuals {zt}. GARCH: Bollerslev[3],introduced GARCH model known as the Generalized auto-regressive conditional heteroskedasticity model which suggest that the timevarying volatility process is a function of both past disturbances and past volatility. The GARCH model is an infinite order ARCH model generated by: q ∑ α ε + ∑β σ 2 i t −i i =1 j =1 sign effect R t = ln [ Pt / Pt −1 ] p i =1 where, z t = ε t σ t is the normalized residual series. The value of g (zt) is a function of both the magnitude and sign of zt and is expressed as: Data source: All data are the daily data obtained from DataStream. In the database, the daily return Rt consisted of daily stock closing price Pt of KLCI, which is measured in local currency. The sample consists of 2869 daily observations on stock returns of the KLCI index. The index spans a period of approximately 11 years from 1/1/1998 to 31/12/2008. In the database, the daily return Rt consisted of daily closing price Pt, which is measured in local currency and computed as: 2 q 2 MATERIALS AND METHODS σt = α0 + p ln σ t = α 0 + ∑ α i g(z t − i ) + ∑ β j ln(σ t − j ) 2 j t− j Normal distribution: (1) j =1 where α0, α and β are non-negative constants. For the GARCH process to be defined, it is required that α>0. f (z) = 235 1 2π z2 , E z = 2 / π , γ2 = 0 2 exp − (5) J. Math. & Stat., 5 (3):234-240, 2009 GED: Skew-normal distribution: 1 − f (z) = e σπ ( z −ζ )2 2 π 4−π 2 γ2 = dt 2π ) 2 (δ ) 1 − 2δ π δ= 1+ α 2 ) Ez = γ2 = ( exp δ τ 2 − β2 + β ( z − µ ) γ1 = z2 1 + v − 2 π ( v − 2 ) 1 (7) = K(u) = , v>4 ( ) ( ) ( ) ( ) = 2 )( ) − 1 ζ + 1 Γ ( v − 3) / 2 4 πζ s Γ ( v / 2 ) 3 3 3( ζ + ζ 5 γ2 (ζ −5 ) ( v − 2 ) − 3m (m 2 ( ζ + ζ ) ( v − 4) s 4m ( v − 2 ) ( ζ − 1)( ζ − −1 4 3/ 2 2 s 4 2 − + 2s) 4 ) + 1 Γ ( v − 3) / 2 πζ s Γ ( v / 2 ) ζ > 0, v > 2, m = Γ ( v − 1) / 2 Γ ( v / 2) , s = (ζ 2 2 ) , ( 3 1 + 4β 2 2∫ ) δω τ= 1 τ2 y λ−1 2 2 ω +β , φ(z) = 1 + ( ( z − µ ) δ ) , u 1 y + dy, u>0 y 2 exp − 2 denotes the Descriptive statistics: Table 1 shows the descriptive statistics of KLCI series for the sample under consideration. The mean return is positive 0.014%, however, accompanied by high volatility 1.51%. It is clear that the Malaysian market offer high average returns but these high returns are also characterized by high volatility, which is common for emerging markets and is consistent with previous studies[9,10]. Moreover, we check the statistical features of the data reported in Table 1, the skewness, kurtosis and their tests. The Ljung-Box Q-statistics Q (10) and Q2 (10) are reported under the null hypothesis of non-serial correlation tests in daily return and squared returns, m(m 2 + 3s) (8) s3 3 4 where, 1/ 2 RESULTS ζ + 1 ζ πΓ ( v 2 )( v − 1) 3/ 2 ) modified Bessel function of the third kind of order λ evaluated at u and w = τ2 − β2 let Ei, T (hT+h) denote the h-step ahead forecast of hT+h at time T from GARCH model i using rolling methods. Define the corresponding forecast error as εI, T+h = Ei, T (hT+h)-hT+h Common evaluation statistics based on Mean Squared Error (MSE), Mean Absolute Error (MAE), Median Squared Error (MDSE) and Root Mean Squared Error (RMSE). 4ζ 2Γ (1 + v) 2 ) v − 2 ( v − 2) ) (10) where, 2s ζ + ζ −1 Γ ( v + 1 2 ) Γ ( v 2 ) 1 π(v − 2) − ( v +1/ 2) 2 2 if z ≤ − m / s 1 + ζ (sz + m) v − 2 f (z) = 2s ζ + ζ −1 Γ ( v + 1 2 ) Γ ( v 2 ) 1 π(v − 2) 1 + ζ −2 (sz + m) 2 v − 2 − ( v +1/ 2) if z > − m / s γ1 τ ( δω) γ2 = 3 + ,v > 2 Skewed-t distribution: Ez = δβ , ω 3β −( v +1 2 ) 4Γ (1 + v / 2 ) v − 2 v−4 v) Γ (5 / v) [Γ ( 3 / v )]2 E z =µ+ 1 + πΓ ( v / 2 )( v − 1) 6 (9) Γ (1 / = ( . Γ ( v + 1 2) Γ ( v 2) Γ (1 / v ) Γ ( 3 / v ) τ −1/ 2 f (z) = φ ( z ) K1 τδ φ(z) π 2 Student-t distribution: f (z; v) = ( −2/ v ) NIG: ξ and α are the location scale and shape respectively, where, ) 4 2 α 2 , 3 2 2π ( γ2 (6) 3 v λ 2(1+1/ v ) Γ (1 / v ) v > 0, λ = (1 − 2δ π) 2 ( π − 3) f ( z; v ) = , (δ v exp ( −0.5 z / λ 2 z −ζ t − σ e 2 α −∞ E z = ζ + σδ γ1 = ∫ σ + 1 / ζ −1 − m 2 ( v − 2 ) / π (ζ − 1 / ζ ) / γ1 exists if v>3 and γ2 exists if v>4. 236 J. Math. & Stat., 5 (3):234-240, 2009 Table 1: Summary statistics for daily returns 1 January 1998-31 December 2008 Sample Mean Std. Skewness Kurtosis Rob.Sk Rob.Kr Q (10) Q2 (10) LM (5) KLCI 2870 0.0135 1.506 0.569 60.239 -0.004 0.37 85.29** 1365** 787** Rob.Sk and Rob.Kr are outlier-robust versions of skewness and kurtosis described as Sk2 and Kr2 in Kim and White[6], LM test of Engle[4] for presence of ARCH at lag 5, **,*: Significant at 1 and 5% respectively -0.2 -0.1 0.0 0.1 0.2 Figure 1 looks at the behavior of the KLCI returns, over the sample period. There is evidence of volatility clustering and that large or small asset price changes tend to be followed by other large or small price changes of either sign (positive or negative). This implies that stock return volatility changes over time. Furthermore, the figure indicates that the Malaysian equity market is not affected much by US recent financial crisis compared to the Asian crisis in 1997. Figure 2 shows the distribution of KLCI log returns, which clearly indicate the departure from normality with a high peaked distribution. 1998-01-02 2000-03-16 2002-05-28 2004-08-09 2006-10-19 Estimation and diagnostic: A quasi maximum likelihood approach is used to estimate the Models GARCH, EGARCH and NAGARCH, with six underlying error distributions. Table 2, shows the estimation results for the parameters for the GARCH, EGARCH and NAGARCH models. The use of asymmetric GARCH models seems to be justified. All asymmetric coefficients are significant at standard levels. Moreover, the Akaike Information Criteria (henceforward AIC) and the Bayesian information criterion values highlight the fact that EGARCH and NAGARCH models better estimate the series than the traditional GARCH. All the models seem to do a good job in describing the dynamic of the first two moments of the series as shown by the Box-Pierce statistics for the squared standardized residuals with lag 15 which are all non-significant at 5% level. LM test for presence of ARCH effects at lag 5, indicate that the conditional hetroskedasity that existed when the test was performed on the pure return series (Table 1) are removed. As, is typical of GARCH model estimates for financial asset returns data, the sum of the coefficients on the lagged squared error (α1) and the lagged conditional variance (β1) is close to unity 1.00 and 0.99 with the GED and NIG error term respectively, this implies that shocks to the conditional variance well be highly persistent indicating that large changes and small changes tend to be followed by small changes, this mean volatility clustering is observed in KLCI financial returns series. 2008-12-31 Fig. 1: KLCI daily returns Fig. 2: KLCI daily return distribution respectively. At significance levels of 5%, the null hypotheses (skewness = 0 or excess kurtosis = 0) and of non-serial correlation are rejected, respectively. This time series have the typical features of stock returns as fat tail, spiked peak and persistence in variance. In contrast, the robustified statistics, Rob.Sk and Rob.Kr, do not suggest non-normality. With evidence of ARCH effects, it is possible to proceed to the second step of the analysis focused on the GARCH modeling of market’s volatility. 237 J. Math. & Stat., 5 (3):234-240, 2009 Table 2: GARCH models estimation Normal Panel A: GARCH: α0 0.0000** (0.0000) 0.1185** α1 (0.0111) 0.8874** β1 (0.0092) γ1 γ2 Log like AIC BIC Q2(15) LM(5) Panel B: EGARCH: α0 α1 β1 φ1 φ2 -9113.6290 -6.3504 -6.3421 8.4750 2.2450 -0.0682** (0.0195) 0.2417 (0.4169) 0.9911** (0.0021) -0.0808* (0.0036) 0.3003* (0.0776) γ1 Skew normal 0.0000** (0.0000) 0.1177** (0.0111) 0.8881** (0.0093) 1.0122** (0.0197) 4.0477** (0.3345) -9113.8230 -6.3498 -6.3394 8.4800 2.2770 -0.0697** (0.0201) 0.4144 (0.5892) 0.9910** (0.0022) -0.1290** (0.0027) 0.4705** (0.0104) 0.9766** (0.0192) γ2 Log like -9155.3020 AIC -6.3773 BIC -6.3628 Q2(15) 9.7570 LM(5) 5.6840 Panel C: NAGARCH: 0.0000** α0 (0.0000) 0.1124** α1 (0.0106) 0.8770** β1 (0.0099) γ 0.3428** (0.0515) γ1 -9156.0020 -6.3771 -6.3605 9.4220 5.3950 0.0000** (0.0000) 0.1132** (0.0108) 0.8757** (0.0103) 0.3475** (0.0523) 0.9873** (0.0195) Student-t 0.0000** (0.00000) 0.14790** (0.02180) 0.86170** (0.01710) 4.04160** (0.33420) -9287.87900 -6.47120 -6.46080 7.64900 1.65710 -0.19600** (0.04680) 0.78050 (0.95070) 0.97860** (0.00510) -0.08660** (0.00180) 0.32060** (0.09270) 4.22540** (0.35380) -9330.24800 -6.49790 -6.47920 5.09100 0.88110 0.00000** (0.00000) 0.14520** (0.02150) 0.84770** (0.01870) 0.31260** (0.06470) γ2 4.11600** (0.34350) Log like -9137.3100 -9137.5190 -9300.47800 AIC -6.3662 -6.3656 -6.47920 BIC -6.3558 -6.3532 -6.46680 Q2(15) 8.8690 8.8380 8.51200 LM(5) 2.6150 2.5790 2.22845 Standard errors are given in parentheses. **,*: Significant at 1 and 5% respectively Skew student 0.0000** (0.00000) 0.14850** (0.02190) 0.86150** (0.01710) 0.98630** (0.02250) 1.02360** (0.05580) -9288.06000 -6.47060 -6.45810 7.65500 1.65970 -0.19850** (0.04790) 0.48790 (0.43130) 0.97830** (0.00520) -0.13830** (0.00130) 0.52040** (0.01810) 0.96360** (0.02310) 4.19210** (0.34980) -9330.62000 -6.49820 -6.47950 4.06400 0.88780 0.00000** (0.00000) 0.14600** (0.02140) 0.84580** (0.01870) 0.32190** (0.06570) 0.97150** (0.02260) 4.11280** (0.34420) -9301.25700 -6.47910 -6.46450 8.46600 2.22186 GED NIG 0.0000** (0.0000) 0.1306** (0.0189) 0.8735** (0.0159) 0.0000** (0.0000) 0.1312** (0.0183) 0.8686** (0.0160) -0.0145 (0.0178) 1.0000** (0.1463) -9301.719 -6.4808 -6.4704 7.786 1.7187 -0.1451** (0.0372) 0.4076 (0.5362) 0.9843** (0.0040) -0.1399** (0.0049) 0.5343** (0.0048) 1.0786** (0.0376) -9328.751 -6.4976 -6.4809 5.773 2.166 0.0000** (0.0000) 0.1287** (0.0183) 0.8588** (0.0173) 0.3503** (0.0696) 1.0341** (0.0407) -9315.2340 -6.4895 -6.4771 8.5310 2.1743 -9290.2330 -6.47210 -6.45960 7.72000 1.66470 -0.1717** (0.0412) 0.37670 (0.49570) 0.981500** (0.00450) -0.16660** (0.00540) 0.61810** (0.01270) -0.05550 (0.04270) 1.00000** (0.13650) -9330.03200 -6.49780 -6.47910 4.69100 1.33330 0.0000** (0.0000) 0.13070** (0.01810) 0.85200** (0.01780) 0.346700** (0.06880) -0.04660 (0.03540) 1.000000** (0.14470) -9304.15000 -6.48110 -6.46660 8.43100 2.14490 perform better with GED, while EGARCH model perform better with skew-student-t distribution. The leverage effect terms φ1 and φ2 in EGARCH model and asymmetric term λ in NAGARCH model are statistically significant, furthermore with φ1 negative sign, DISCUSSION Based on the findings, the symmetric distributions with fatter tails clearly outperform the Gaussian. According to AIC, GARCH and NAGARCH models 238 J. Math. & Stat., 5 (3):234-240, 2009 Table 3: Forecast performance out-of- sample Normal Skew normal Student-t Skew student GED Panel A: GARCH: MSE 0.001098 0.001098 0.001294 0.001313 0.001099 MDSE 0.000212 0.000212 0.000351 0.000362 0.000219 RMSE 3.313204 3.313204 3.597198 3.623870 3.314653 MAE 2.143484 2.143484 2.649073 2.683303 2.155249 Panel B: EGARCH: MSE 0.001092 0.001072 0.001090 0.001090 0.001096 MDSE 0.000339 0.000339 0.000118 0.000115 0.000107 RMSE 3.304849 3.274849 3.300770 3.301175 3.309850 MAE 2.244085 2.244085 1.851566 1.852097 1.838961 Panel C: NAGARCH: MSE 0.002087 0.002087 0.013221 0.011741 0.003112 MDSE 0.000908 0.000908 0.003131 0.002976 0.001271 RMSE 4.568193 4.568193 11.498303 10.835615 5.578780 MAE 3.645259 3.645259 8.491675 8.080493 4.479974 MSE: Mean squared error; MDSE: The median squared error; RMSE: The root mean squared error; MAE: The mean absolute error NIG 0.001080 0.000159 3.286948 2.008332 0.001100 0.000100 3.316831 1.833950 0.001978 0.000850 4.447064 3.552767 residuals (i.e., NIG distribution). A possible explanation is that modeling asymmetries contributes to the reduction of the magnitude of the bias. as expected that negative shocks imply a higher next period conditional variance than positive shocks of the same sign, indicating that the existence of leverage effect is observed in returns of the KLCI market index. However, the comparison between models with each density (normal versus non-normal) shows that, according to the different measures used for modeling the volatility, the EGARCH model with skew-student-t provides the best in-sample estimation for KLCI compared to all other volatility models and distributions. Now that we have estimated the series, the obvious question is how good are the forecasting models? Typically, there are several plausible models that we can select to use for our forecast. We should not be fooled into thinking that the one with the best fit is the one that will forecast the best. The results of forecasting daily volatility with GARCH models together with various distributions and four evaluation criteria are given in Table 3. All results are presented for each distribution and for each model GARCH models, whose specification is always of order (1, 1). In this study the length of the out-of-sample period is chosen to be 100 days. Form Table 3 some interesting comments emerge. A first major conclusion is that there is no single model that completely dominates the other models. Secondly, forecasting with normal distribution does not yield a significant reduction of the forecast error relative to heavy-tailed distributions. Thus, the failure of predictor hT+h is justified due to that the GARCH model residuals follow a (possibly) heavy-tailed distribution. Third, it seems that asymmetric model (EGARCH) tend to perform better forecast with a fatter tailed distribution (student-t and Skew-student-t). It is apparent, that the simple predictor hT+h seems to actually have some predictive ability, when a heavy-tailed is assumed for the GARCH CONCLUSION This study contributes to the literature of volatility modeling in two aspects. First, we use a data set from an emerging market. Secondly we estimate the alternative GARCH-type models (symmetric and asymmetric GARCH Models). The comparison was focused on two different aspects: The difference between symmetric and asymmetric GARCH (i.e., GARCH versus EGARCH and NAGARCH) and the difference between normal tailed symmetric, heavytailed symmetric distributions and both heavy-tailed and asymmetric distributions for estimating the KLCI stock market index return volatility. As expected, the leverage KLCI market shown by EGARCH model is statistically significant at with a negative sign, which indicate that negative shocks imply a higher next period conditional variance than positive shocks of the same sign, indicating that the existence of leverage effect is observed in returns of the KLCI stock market index. However, the comparison between models with each density (normal versus non-normal) shows that, according to the different measures used for the performance of volatility forecast, the EGARCH model provides the best out-sample estimation for KLCI clearly outperform the symmetric models. Our results show that, non-normal distributions provide better insample results than the normal distribution. Out-ofsample results show however less evidence of superior forecasting ability. 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