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 R = 2000 observations, while the last P = 1000 observations are used for the out-of-sample forecast. Only the case of generating one-step ahead forecasts using the three alternative methods to generate a sequence of P one-step ahead forecasts is considered. For the estimation sample sizes R for all return series, the study will consider five different values for P for the three alternative schemes, namely P = 200, 400, 600, 800, 1000.

shocks (either negative or positive) to (skewness & kurtosis) or to lagged (skewness & kurtosis)

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.

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 zt in both specifications. The results are contained in Table 7. The value of the LR statistic is quite large in all return series. This means that the GARCHSK model is showing superior performance than

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\*

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

are found to be significant.

50 Time Series Analysis and Applications

3. Comparison of models

Stocks

Exchange rates

\*

4. A new forecast scheme

Significant at the 5% level.

the standard GARCH model with constant shape parameters.

Table 7. Likelihood ratio tests for all daily returns of stock and exchange rate series.

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.

Table 8 presents the averages of the mean square forecasting errors over all levels of out-ofsample forecast (P = 200, 400, 600, 800, 1000) for the recursive, rolling, fixed, and weighted average schemes for all daily returns of stock and exchange rate series. The results show that the average forecasting mean squares errors for the four forecasting methods for all return series differ only either in the second decimal place or third decimal place. Although the weighted method shows clear superiority to the recursive and fixed methods, it failed to beat the rolling method which outperforms all other three methods in these data. We attribute the fair performance of weighted method compared to the rolling method possibly because of the


Table 8. Averages of the mean square forecasting errors over all levels of out-of-sample forecast (P = 200, 400, 600, 800, 1000) for all forecasting alternative schemes for all daily returns of stock and exchange rate series.

small differences in the mean square errors of the un-weighted methods. We expect it to perform better in cases, where the three methods differ markedly with respect to their mean square errors.
