Normal Log-normal Mixture: Leptokurtosis, Skewness and Applications

The properties and applications of the normal log-normal (NLN) mixture are considered. The moment of the NLN mixture is shown to be finite for any positive order. The expectations of exponential functions of a NLN mixture variable are also investigated. The kurtosis and skewness of the NLN mixture are explicitly shown to be determined by the variance of the log-normal and the correlation between the normal and log-normal. The issue of testing the NLN mixture is discussed. The NLN mixture is fitted to a set of cross-sectional data and a set of time-series data to demonstrate its applications. In the time series application, the ARCH-M effect and leverage effect are separately estimated and both appear to be supported by the data