Normal log-normal mixture, leptokurtosis and skewness

Abstract

The normal log-normal mixture (NLNM) is considered for modelling leptokurtosis and skewness. The moment of the mixture is shown to be finite for any positive order. The expectations of exponential functions of the NLNM variable are also investigated. The kurtosis and skewness of the NLNM are explicitly shown to be determined by the variance of the log normal and the correlation between the normal and log normal. A set of cross-sectional data is fitted to the NLNM to illustrate that idiosyncratic variances may be modelled by the mixture.

Notes

¹Testing M0 directly against M2 is not attempted here since ρ is unidentified under M0 and the test is involved (Andrews and Ploberger, 1994).