Descriptive statistics for daily returns.
This paper proposes a GARCH-type model allowing for time-varying 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 time-varying 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 time-varying 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.
- GARCH models
- conditional volatility
- skewness and kurtosis
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 left-hand 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  and Bollerslev , allow for time-varying 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” .
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” .
This paper contributes to the literature of volatility modeling in two aspects. First, we jointly estimate time-varying 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 US-dollar 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).
Some of the closing price indices were put into US-dollar 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).
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 Ljung-Box statistics of order 20 for
|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
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, ARCH-type 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 time-varying 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 time-varying shape parameters.
ARCD is the approach, where dynamics imposed on shape parameters and skewness or kurtosis are derived from the time-varying shape parameters.
For stock return series:
Variance equation (GARCH)
For all stock return series, the study is going to use GARCH(1,1) model with a similar specification to that of Hansen  for shape parameters (
For exchange rate return series:
Variance equation (GARCH)
For the exchange rate return series, a specification similar to that of  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 (
|ARCH-LM test for heteroscedasticity|
|Prob. chi-square (5)||0.2525||0.1691||0.9880||0.9259||0.5447|
|ARCH-LM test for heteroscedasticity|
|Prob. chi-square (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.
|ARCH-LM test for heteroscedasticity|
|Prob. chi-square (5)||0.0754**||0.7155||0.1340**||0.0859||0.0016|
|ARCH-LM test for heteroscedasticity|
|Prob. chi-square (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 ARCH-LM 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|
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 . 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|
Table 8 presents the averages of the mean square forecasting errors over all levels of out-of-sample forecast (
This chapter proposes a GARCH-type model that allowing for time-varying 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 time-varying 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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- 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” .
- Some of the closing price indices were put into US-dollar 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 time-varying shape parameters.