Bayesian MCMC analysis of periodic asymmetric power GARCH models

A Bayesian MCMC estimate of a periodic asymmetric power GARCH (PAP-GARCH) model whose coefficients, power, and innovation distribution are periodic over time is proposed. The properties of the PAP-GARCH model such as periodic ergodicity, finiteness of moments and tail behaviors of the marginal distributions are first examined. Then, a Bayesian MCMC estimate based on Griddy-Gibbs sampling is proposed when the distribution of the innovation of the model is standard Gaussian or standardized Student with a periodic degree of freedom. Selecting the orders and the period of the PAP-GARCH model is carried out via the Deviance Information Criterion (DIC). The performance of the proposed Griddy-Gibbs estimate is evaluated through simulated and real data. In particular, applications to Bayesian volatility forecasting and Value-at-Risk estimation for daily returns on the S&P500 index are considered.