Pseudo-variance quasi-maximum likelihood estimation of semi-parametric time series models

TL;DR

This paper introduces a new pseudo-variance quasi-maximum likelihood estimation method for semi-parametric time series models, enhancing efficiency and providing a framework for testing model specifications, applicable to bounded and count data.

Contribution

It develops a novel estimation approach based on Gaussian quasi-likelihood with pseudo-variance, applicable to observation-driven models with support bounds, and derives their asymptotic properties.

Findings

Estimates remain valid regardless of pseudo-variance specification.

Restricted estimators achieve higher efficiency than existing methods.

Method successfully applied to empirical count and bounded data models.

Abstract

We propose a novel estimation approach for a general class of semi-parametric time series models where the conditional expectation is modeled through a parametric function. The proposed class of estimators is based on a Gaussian quasi-likelihood function and it relies on the specification of a parametric pseudo-variance that can contain parametric restrictions with respect to the conditional expectation. The specification of the pseudo-variance and the parametric restrictions follow naturally in observation-driven models with bounds in the support of the observable process, such as count processes and double-bounded time series. We derive the asymptotic properties of the estimators and a validity test for the parameter restrictions. We show that the results remain valid irrespective of the correct specification of the pseudo-variance. The key advantage of the restricted estimators is that they can achieve higher efficiency compared to alternative quasi-likelihood methods that are available in the literature. Furthermore, the testing approach can be used to build specification tests for parametric time series models. We illustrate the practical use of the methodology in a simulation study and two empirical applications featuring integer-valued autoregressive processes, where assumptions on the dispersion of the thinning operator are formally tested, and autoregressions for double-bounded data with application to a realized correlation time series.