Descriptive statistics for daily returns.
Abstract
This paper proposes a GARCHtype model allowing for timevarying volatility, skewness, and kurtosis assuming a Johnson’s SU distribution for the error term. This distribution has two shape parameters and allows a wide range of skewness and kurtosis. We then impose dynamics on both shape parameters to obtain autoregressive conditional density (ARCD) models, allowing timevarying skewness and kurtosis. ARCD models with this distribution are applied to the daily returns of a variety of stock indices and exchange rates. Models with timevarying shape parameters are found to give better fit than models with constant shape parameters. Also, a weighted forecasting scheme is introduced to generate the sequence of the forecasts by computing a weighted average of the three alternative methods suggested in the literature. The results showed that the weighted average scheme did not show clear superiority to the other three methods.
Keywords
 GARCH models
 conditional volatility
 skewness and kurtosis
1. Introduction
Many papers deal with the departures from normality of asset return distributions. It is well known that the distributions of stock return exhibit negative skewness and excess kurtosis; see among others [2, 9, 14, 15]. The higher moments of the return specifically, excess kurtosis (the fourth moment of the distribution) makes extreme observations more likely than in the normal case, which means that the market gives higher probability to extreme observations than in normal distribution. However, the existence of negative skewness (the third moment of the distribution) has the effect of accentuating the lefthand side of the distribution, which means that a higher probability of decreases given to asset pricing than increases in the market.
The generalized autoregressive conditional heteroscedasticity (GARCH) models, introduced by Engle [5] and Bollerslev [1], allow for timevarying volatility
In general terms, volatility refers to the fluctuations observed in some phenomenon overtime. In terms of modeling and forecasting literature, it means “the conditional variance of the underlying asset return” [17].
but not for timevarying skewness or timevarying kurtosis. Different GARCH models have been developed in the literature to capture dependencies in higher order moments, starting with Hansen [7] who proposed a skewStudent distribution to account for both timevarying excess kurtosis and skewness. A significant evidence of timevarying skewness found [9]. Others [11, 12] found a significant time varying in both skewness and kurtosis, while [3, 15, 16] found little evidence of either. With regard to the frequency of observation, Jondeau and Rockinger [11] found the presence of timevarying skewness and kurtosis in daily but not weekly data, while others including [2, 7, 9] found an evidence of timevarying skewness and kurtosis in weekly and even monthly data. Regarding daily data [4, 12, 18] found an evidence of timevarying skewness and kurtosis in daily data. The chapter employed GARCH(1,1) model as the performance of the model proved compared large number of volatility models; for more details, see Hansen and Lunde [8].This paper contributes to the literature of volatility modeling in two aspects. First, we jointly estimate timevarying volatility, skewness, and kurtosis assuming Johnson SU distribution for the error term. The method is applied to two different daily returns: stock indices and exchange rates. Second, a new alternative scheme is introduced to generate the sequence of the forecasts.
The rest of the paper is organized as follows. Following this introduction, Section 2 presents the empirical results regarding the estimation of the model. Section 3 compares the models. In Section 4, the new forecasting scheme is presented, while Section 5 gives concluding remarks.
2. Empirical results and methodology
2.1. Data and preliminary findings
The time series data used for modeling volatility in this paper consists of two sets of financial data. The first set includes daily returns of five stock indices: NASDAQ100 (US), Germany (DAX30), Ishares MSCI South Africa index (EZA), Shanghai stock exchange composite index (SSE), and Ishares MSCI Canada index (EWC).
Some of the closing price indices were put into USdollar and some were put into other currencies. For unification of foreign exchange rates, all closing price indices were converted into American US dollar. These closing price indices are obtained from Yahoo Finance (http://finance.yahoo.com).
The second data set includes daily returns of five exchange rates series: British Pound (USD/GBP), Australian Dollar (USD/AUD), Italian Lira (USD/ITL), South Africa Rand (USD/ZAR), and Brazilian Real (USD/BRL).The exchange rates have been retrieved from the website (http://www.oanda.com).
Assets  N  Mean  S.D.  Skewness  Kurtosis  JarqueBera 

NASDAQ100  2000  0.011  1.789  0.084  7.139  1429.85* 
DAX30  2000  0.032  1.795  0.053  6.473  1929.78* 
SSE  2000  0.048  1.764  −0.078  6.929  1292.92* 
EZA  2000  0.076  2.403  −0.354  14.436  10968.85* 
EWC  2000  0.049  1.673  −0.473  9.327  3420.18* 
USD/GBP  2000  0.007  0.485  0.658  11.419  6066.76* 
USD/AUD  2000  −0.013  0.702  0.481  14.254  10659.08* 
USD/ITL  2000  −0.004  0.467  −0.197  8.185  2260.57* 
USD/ZAR  2000  0.001  0.877  1.010  17.404  17672.41* 
USD/BRL  2000  −0.016  0.961  0.441  10.048  4215.97* 
2.2. Methodology
Preliminary results in the preceding section provided evidence of a significant deviation from normality and obvious leptokurtosis in all daily return series. This suggests specifying GARCH models that capture these characteristics. In presenting these models, there are two distinct equations or specifications, one for the conditional mean and the other for the conditional variance. For the models employed in this paper, the mean equation for all stock return series is the AR(1) model with a constant, and for all exchange rate return series, we used the MA(1) model without a constant. After estimating the mean equation, the next step was to identify whether there is substantial evidence of heteroscedasticity for the daily returns of stock and exchange rate series. Table 2 provides the LjungBox statistics of order 20 for
Series 




NASDAQ100  1834.3 (0.000)  305.1 (0.000)  507.1 (0.000) 
DAX30  2132.9 (0.000)  148.4 (0.000)  676.1 (0.000) 
SSE  443.2 (0.000)  24.6 (0.216)  52.4 (0.000) 
EZA  2597.2 (0.000)  305.8 (0.000)  647.8 (0.000) 
EWC  3614.3 (0.000)  272.1 (0.000)  984.2 (0.000) 
USD/GBP  1020.8 (0.000)  98.6 (0.000)  190.6 (0.000) 
USD/AUD  2525.9 (0.000)  678.2 (0.000)  889.8 (0.000) 
USD/ZAR  975.5 (0.000)  89.2 (0.000)  39.128 (0.006) 
USD/ITL  536.2 (0.000)  94.477 (0.000)  77.6 (0.000) 
USD/BRL  1555.3 (0.000)  406.1 (0.000)  1030.9 (0.000) 
2.2.1. Distributional assumptions
To complete the basic GARCH specification, an assumption about the conditional distribution of the error term
where
where
The quantities Ω and ω in the moment formulas are Ω = γ/δ and ω = exp(δ−2). The skewness and kurtosis are jointly determined by the two shape parameters γ and δ. The standardized Johnson’s SU innovations exist when ξ = 0 and λ = 1, but the mean and the variance are not 0 and 1, respectively. These can be done by setting the parameters in the following manner:
2.2.2. Maximum likelihood
Under the presence of heteroscedasticity (autoregressive conditional heteroscedasticity (ARCH) effects) in the residuals of the daily returns of stock and exchange rate series, the ordinary least square estimation (OLS) is not efficient, and the estimate of covariance matrix of the parameters will be biased due to invalid ‘t’ statistics. Therefore, ARCHtype models cannot be estimated by simple techniques such as OLS. The method of maximum likelihood estimation is employed in ARCH models. For the formal exposition of the approach, each realization of the conditional variance
The log likelihood function is:
The parameter values are selected so that the log likelihood function is maximized using a search algorithm by computers.
2.2.3. Model estimation with timevarying volatility, skewness, and kurtosis
As it was shown in Section 2.2, when the residuals were examined for heteroscedasticity, the Ljung Box test provided strong evidence of ARCH effects in the residuals series, which suggests proceeds with modeling the returns volatility using the GARCH methodology. The model to be estimated in this study is the standard GARCH(1, 1) model with constant shape parameters, and also, we impose dynamics on both shape parameters to obtain autoregressive conditional density (ARCD) models.
ARCD is the approach, where dynamics imposed on shape parameters and skewness or kurtosis are derived from the timevarying shape parameters.
This allows for timevarying skewness and kurtosis assuming Johnson Su distribution for the error term in the two cases. Before presenting the estimation results obtained with both the stock return series and the exchange rate return series, the four nested models to be estimated are summarized as follows:For stock return series:
Mean equation
Variance equation (GARCH)
Skewness equation
Kurtosis equation
For all stock return series, the study is going to use GARCH(1,1) model with a similar specification to that of Hansen [7] for shape parameters (
For exchange rate return series:
Mean equation
Variance equation (GARCH)
Skewness equation
Kurtosis equation
For the exchange rate return series, a specification similar to that of [11] for shape parameters (
The results for the stock return series are presented in Tables 3 and 4 for both the standard GARCH and GARCHSK models, respectively. As expected, the results indicate high and significant presence of conditional variance, since the coefficient of lagged conditional variance (
Parameters  NASDAQ100  DAX30  SSE  EZA  EWC  

Mean equation  0.0536*  0.0940*  0.0207  0.1535*  0.0976*  
−0.0578*  −0.0813*  0.0025  −0.0534*  −0.0461*  
Variance equation  0.0082  0.0128*  0.0284*  0.0596*  0.0202*  
0.0499*  0.0646*  0.0756  0.1011*  0.0619*  
0.9468*  0.9311*  0.9225*  0.8894*  0.9285*  
Loglikelihood  −3589.94  −3588.5  −3651.1  −4178.55  −3308.61  
AIC  3.5969  3.5955  3.6580  4.1855  4.1445  
ARCHLM test for heteroscedasticity  
Statistic ( 
6.596  7.775  0.5993  1.385  4.032  
Prob. chisquare (5)  0.2525  0.1691  0.9880  0.9259  0.5447 
Parameters  NASDAQ100  DAX30  SSE  EZA  EWC  

Mean equation  0.0155  0.0816*  0.0555  0.1312*  0.0851*  
−0.0567*  −0.0947*  −0.0154  −0.0512*  −0.0540*  
Variance equation  0.0104*  0.0167*  0.0506*  0.0620*  0.0250*  
0.0578*  0.0717*  0.1009*  0.0931*  0.0762*  
0.9436*  0.9239*  0.8997*  0.8998*  0.9183*  
Skewness equation  −0.0038*  0.0035*  0.0015*  −0.0261*  −0.0256*  
0.00002  −0.0083*  −0.0054*  0.0838*  0.0163  
0.00355*  −0.0037*  −0.0017*  0.0004  0.0192*  
0.9939*  1.0000*  0.9898*  0.8661*  0.9165*  
Kurtosis equation  0.0001  0.7193*  0.9625*  0.2245*  0.4362  
0.9869*  0.3126*  0.2684*  0.4848*  0.5166*  
0.0799  0.2929*  0.0591  0.0000  0.2638*  
0.8459*  0.0019  0.5469*  0.8143*  0.4358*  
Loglikelihood  −3559.79  −3578.15  −3620.83  −3294.5  −3406.96  
AIC  3.5728  3.5911  3.6338  4.1344  3.4200  
ARCHLM test for heteroscedasticity  
Statistic ( 
6.942  6.604  1.678  0.7606  5.393  
Prob. chisquare (5)  0.2250  0.2518  0.8917  0.9795  0.3698 
The results for the five exchange rates are presented in Tables 5 and 6 for GARCH and GARCHSK models, respectively. As expected, the results are the same as in the case of stock return series, i.e., the results also indicate highest significant presence of conditional variance. Volatility is found to be persistent, and volatility clustering is also observed in exchange rate return series. A significant presence of conditional skewness and kurtosis for all exchange rate return series is confirmed, since at least one of the coefficients associated with the standardized shocks (either negative or positive) to (skewness & kurtosis) or to lagged (skewness & kurtosis) are found to be significant.
Parameters  USD/GBP  USD/AUD  USD/ITL  USD/ZAR  USD/BRL  

Mean equation  0.28470*  0.1886*  0.2495*  0.2619*  0.0945*  
Variance equation  0.0009*  0.0015*  0.0006  0.0165*  0.0114  
0.0384*  0.0485*  0.0331*  0.0553*  0.1041  
0.9579*  0.9505*  0.9658*  0.9175*  0.8948*  
Loglikelihood  −907.732  −1528.337  −922.161  −2257.187  −2159.827  
AIC  0.9137  1.5343  0.9282  2.2632  2.1658  
ARCHLM test for heteroscedasticity  
Statistic ( 
5.169  2.900  4.019  9.646  28.35  
Prob. chisquare (5)  0.0754**  0.7155  0.1340**  0.0859  0.0016 
Parameters  USD/GBP  USD/AUD  USD/ITL  USD/ZAR  USD/BRL  

Mean equation  0.2978*  0.2111*  0.2626*  0.2590*  0.0978*  
Variance equation  0.0009  0.0016  0.0006  0.0139*  0.0086*  
0.0502*  0.0597*  0.0425*  0.0760*  0.2626*  
0.9489*  0.9449*  0.9582*  0.9119*  0.8348*  
Skewness equation  −0.0306  0.0368*  −0.0189  0.0168*  −0.0047  
0.0237  0.0610*  0.0195  0.0589*  −0.0051  
0.0808*  0.0036  0.0658*  0.0058  0.0150*  
0.0000  0.4814  0.0000  0.9018*  0.8807*  
Kurtosis equation  0.2075  0.2939*  0.2128  0.4497  0.0405  
0.4029*  0.5678*  0.3459*  1.0000*  1.0000*  
0.0050  0.0000  0.0235  0.0000  0.0000  
0.8217*  0.7851*  0.8364*  0.5342*  0.9077*  
Loglikelihood  −895.695  −1516.323  −910.919  −2227.667  −2135.46  
AIC  0.9077  1.5283  0.9229  2.2397  2.1475  
ARCHLM test for heteroscedasticity  
Statistic ( 
4.299  2.4075  3.308  8.659  9.116  
Prob. chisquare (5)  0.1165  0.7904  0.1912^{**}  0.1235  0.1045 
Finally, it is worth noting that from the bottom of Tables 3–6, the value of Akaike information criterion (AIC) decreases monotonically when moving from the simpler model (standard GARCH) to the more complicated ones (GARCHSK) for all return series. Therefore, for all return series analyzed, the GARCHSK model specification seems to be the most appropriate one according to the AIC. Note that the ARCHLM test statistics for all return series did not exhibit additional ARCH effect. This shows that the variance equations are well specified and adequate.
3. Comparison of models
One way to start comparing the models is to compute the likelihood ratio test. The LR test statistic has been used to compare the standard GARCH model (restricted model) and GARCHSK model (unrestricted model), where Johnson Su distribution is assumed for the standardized error
Series  LogL (GARCH)  LogL (GARCHSK)  LR 

NASDAQ100  −3589.94  −3559.79  60.3* 
DAX30  −3588.5  −3578.15  20.7* 
SSE  −3651.1  −3620.83  60.54* 
EZA  −3308.61  −3294.5  28.22* 
EWC  −3415.2  −3406.96  16.48* 
USD/GBP  −907.732  −895.695  24.07* 
USD/AUD  −1528.337  −1516.323  24.03* 
USD/ITL  −922.161  −910.919  22.48* 
USD/ZAR  −2257.187  −2227.667  59.04* 
USD/BRL  −2159.827  −2135.46  48.73* 
4. A new forecast scheme
In the literature, three alternative ways for generating the sequence of the forecasts, namely the recursive, rolling, and fixed schemes are suggested, see [13]. In this paper, the estimation sample of the models for all return series is based on
In this section, an attempt is made to introduce a new alternative scheme to generate the sequence of the forecasts by computing a weighted average of the last three alternative methods. The weights used are the reciprocals of the MSE of the methods. The rationale behind this is that a method with large mean square forecasting errors (MSE) (i.e., less reliability) should be given a smaller weight. The suggested name for the new method is “weighted average scheme.” The four forecasting alternative schemes are applied using the estimated GARCHSK models for stock and exchange rate return series, which are given in the previous section and the results are shown in Table 8.
Forecasting alternative schemes  

Series  Recursive  Rolling  Fixed  Weighted 
NASDAQ100  1.521857  1.522096  1.522586  1.522166 
DAX30  2.256312  2.238891  2.254930  2.249675 
SSE  1.736101  1.736698  1.736048  1.736175 
EZA  3.759198  3.752719  3.759654  3.756829 
EWC  2.031167  2.027740  2.031093  2.029841 
USD/GBP  0.093255  0.092812  0.092784  0.092932 
USD/AUD  0.255625  0.255306  0.255633  0.255505 
USD/ITL  0.178520  0.178018  0.178496  0.178318 
USD/ZAR  0.491262  0.489874  0.491256  0.490684 
USD/BRL  0.377914  0.376564  0.377805  0.377420 
Table 8 presents the averages of the mean square forecasting errors over all levels of outofsample forecast (
5. Conclusions
This chapter proposes a GARCHtype model that allowing for timevarying volatility, skewness, and kurtosis where assuming a Johnson’s SU distribution for the error term. Models estimated using daily returns of five stock indices and five exchange rate series. The results indicate significant presence of conditional volatility, skewness, and kurtosis. Moreover, it is found that specifications allowing for timevarying skewness and kurtosis outperform specifications with constant third and fourth moments. Also, a weighted average forecasting scheme is introduced to generate the sequence of the forecasts by computing a weighted average of the three alternative methods namely the recursive, rolling, and fixed schemes are suggested. The results showed that the weighted average scheme did not show clear superiority to the other three methods. Further work will consider linear and nonlinear combining methods and different forecasting horizons to forecast stock and return series.
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Notes
 In general terms, volatility refers to the fluctuations observed in some phenomenon overtime. In terms of modeling and forecasting literature, it means “the conditional variance of the underlying asset return” [17].
 Some of the closing price indices were put into USdollar and some were put into other currencies. For unification of foreign exchange rates, all closing price indices were converted into American US dollar. These closing price indices are obtained from Yahoo Finance (http://finance.yahoo.com).
 The exchange rates have been retrieved from the website (http://www.oanda.com).
 ARCD is the approach, where dynamics imposed on shape parameters and skewness or kurtosis are derived from the timevarying shape parameters.