In this paper, the S&P 500 stock index is studied for its time varying volatility and 
stylized facts. The ARMA mean equation with asymmetric power ARCH errors is 
used to model the series correlations and the conditional heteroscadesticity in the asset 
returns. The conditional distributions of the standardized residuals are assumed to be 
the normal distribution, the t distribution or the skew-t distribution. Furthermore, to 
capture the asymmetry and fat tail of the returns, an ARMA (0, 2)-APARCH (1, 1) 
model with the skew-t distribution is found to fit the data better than other models 
discussed in this paper. Finally, we use ARMA (0, 2)-APARCH (1, 1) model with the 
skew-t distribution to do the 10-step-ahead forecasting compared with the 
ARMA (0, 2)-GARCH (1, 1) model with the normal distribution and get the empirical 
conclusion that the ARMA (0, 2)-APARCH (1, 1) with the skew-t distribution also 
gives a better result in the forecasting.

Modeling S&P 500 STOCK INDEX Using ARMA-ASYMMETRIC POWER ARCH Models PDF

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