ECM Algorithm for Auto-Regressive Multivariate Skewed Variance Gamma Model with Unbounded Density

Abstract

The multivariate skewed variance gamma (MSVG) distribution is useful in modelling data with high density around the location parameter along with moderate heavy-tailedness. However, the density can be unbounded for certain choices of shape parameter. We propose a modification to the expectation-conditional maximisation (ECM) algorithm to calculate the maximum likelihood estimate (MLE) by introducing a small region to cap the conditional expectations in order to deal with the unbounded density. To facilitate application to financial time series, the mean is further extended to include autoregressive terms. Finally, the MSVG model is applied to analyse the returns of five daily closing price market indices. Standard error (SE) for the estimated parameters are computed using Louis’ method.