Quasi-maximum likelihood estimation of periodic GARCH processes

TL;DR

This paper proves the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator for periodic GARCH models with time-varying parameters, under minimal moment conditions.

Contribution

It establishes theoretical properties of QMLE for P-GARCH processes, including existence of stationary solutions and finite moments, which were previously unaddressed.

Findings

QMLE is strongly consistent for P-GARCH models.

QMLE is asymptotically normal under minimal conditions.

Conditions for the existence of stationary solutions are provided.

Abstract

This paper establishes the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for a GARCH process with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution for the periodic GARCH (P-GARCH) equation. As a result, it is shown that the moment of some positive order of the P-GARCH solution is finite, under which we prove the strong consistency and asymptotic normality (CAN) of the QMLE without any condition on the moments of the underlying process.