Quasi‐maximum likelihood estimation of periodic GARCH and periodic ARMA‐GARCH processes

Abstract. This article establishes the strong consistency and asymptotic normality (CAN) of the quasi‐maximum likelihood estimator (QMLE) for generalized autoregressive conditionally heteroscedastic (GARCH) and autoregressive moving‐average (ARMA)‐GARCH processes with periodically time‐varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution of the periodic GARCH (PGARCH) equation. As a result, it is shown that the moment of some positive order of the PGARCH solution is finite, under which we prove the strong consistency and asymptotic normality of the QMLE for a PGARCH process without any condition on its moments and for a periodic ARMA‐GARCH (PARMA‐PGARCH) under mild conditions.