Estimation and Asymptotic Properties in Periodic GARCH(1, 1) Models

This article studies the probabilistic structure and asymptotic inference of the first-order periodic generalized autoregressive conditional heteroscedasticity (PGARCH(1, 1)) models in which the parameters in volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Second, using the representation of squared PGARCH(1, 1) model as a PARMA(1, 1) model, we then consider Yule-Walker type estimators for the parameters in PGARCH(1, 1) model and derives their consistency and asymptotic normality. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases can be more efficient than the least squares estimate (LSE). We use a residual bootstrap to define bootstrap estimators for the Yule-Walker estimates and prove the consistency of this bootstrap method. A set of numerical experiments illustrates the practical relevance of our theoretical results.