Abstract
This investigation proposes a composite Simpson's rule, a numerical integral method, for estimating quantiles on the skewed generalized error distribution (SGED). Daily spot prices of S&P500 and Dow-Jones stock indices are used as data to examine the one-day-ahead VaR (Value at Risk) forecasting performance of the GARCH-N and GARCH-SGED models. Empirical results show that the GARCH-SGED models provide more accurate VaR forecasts than the GARCH-N models for both low and high confidence levels. These findings demonstrate that the use of SGED distribution, which explicitly accommodates both skewness and kurtosis, is essential for out-of-sample VaR forecasting in US stock markets.
Notes
1For instance, the Asian financial crisis in 1997, near bankruptcy of Long Term Capital Management in 1998, bankruptcy of Baring's Bank in 1995, and so on.
2VaR has become an essential tool within financial markets, which refers to the potential portfolio loss over a pre-determined period at a given confidence level.
3Another analogous skewed student-t density was proposed by Hansen (Citation1994).
4The general consensus regarding volatility forecasting in most of the literature is that generalized autoregressive conditional heteroskedasticity (GARCH) models. This study thus, considers the applicability of the GARCH (1,1) model in modelling VaRs.
5Models are estimated with daily trading data for a total of 1906 observations. The estimation period is then rolled forward by adding one new day and dropping the most distant day. This procedure prevents overlap and ensures a fixed size for the sample size used in the model estimation. Thus, there are 250 one-day-ahead VaR forecasts for each of the return series.
6The parameters are estimated by QMLE (Quasi maximum likelihood estimation) and the BFGS optimization algorithm, using the econometric package of WinRATS 6.1.
7See Jeffreys and Jeffreys (Citation1988) and Horwitz (Citation2001) for more details.