Handy sufficient conditions for the convergence of the maximum likelihood estimator in observation-driven models

TL;DR

This paper provides general, easy-to-verify conditions ensuring the convergence and consistency of the maximum likelihood estimator in a broad class of observation-driven time series models, including NBIN-GARCH and NM-GARCH.

Contribution

It extends previous results by allowing the conditional law to depend on the parameter and establishes ergodic solutions and MLE consistency under simple conditions.

Findings

Conditions apply to a wide class of models

Proves consistency of MLE for NBIN-GARCH and NM-GARCH

Demonstrates ergodic solutions exist under these conditions

Abstract

This paper generalizes asymptotic properties obtained in the observation-driven times series models considered by \cite{dou:kou:mou:2013} in the sense that the conditional law of each observation is also permitted to depend on the parameter. The existence of ergodic solutions and the consistency of the Maximum Likelihood Estimator (MLE) are derived under easy-to-check conditions. The obtained conditions appear to apply for a wide class of models. We illustrate our results with specific observation-driven times series, including the recently introduced NBIN-GARCH and NM-GARCH models, demonstrating the consistency of the MLE for these two models.