Stochastic dynamics in financial markets: An empirical investigation

Two well documented empirical regularities in asset markets, leptokurtosis and clustered volatility, are essential for logically consistent and correct financial models as well as exact statistical tests. These regularities also have serious implications for the capital asset pricing model, the option pricing model and the efficient market hypothesis. The GARCH process essentially models time-varying variance or volatility clustering. To determine the exact form of error distribution, and hence the correct probability distribution of stock price changes, this study examined the GARCH model under two distributional assumptions: the normal and the power exponential. The GARCH process was further extended to model factors violating the iid assumption--autocorrelation, the day-of-the-week and seasonality. The study extends previous research in two specific ways. One, it estimates for the first time a GARCH model with a power exponential error distribution that accounts for the day-of-the week and seasonality effects in both the mean and variance. Two, the study employs more recent data. The GARCH models are estimated, for the most part, by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm with a cubic or quadratic (STEPBT) line search method within the framework of the Constrained Maximum Likelihood (CML) Gauss module. The evidence shows that the GARCH-pe process fits the sample data better than the GARCH-n process. Nonlinear dependence in the raw data was not removed in all cases, although linear dependence and observed leptokurtosis were reduced significantly. Unfortunately, both GARCH processes were rejected for all indices. Different volatility in returns for each day of the week is the most prevalent of all the market anomalies. The weekend effect on variance is confirmed for NASDAQ. Low variances on Wednesdays are not supported. Seasonal volatility in returns is considerable. Different mean returns for each day of the week is not a widespread phenomenon. The weekend effect on the mean is weak and only for NASDAQ. Higher Wednesday returns find limited support. Seasonal differences in mean returns is detected in only a few cases.