Theory and Inference for a Class of Observation-driven Models with Application to Time Series of Counts

TL;DR

This paper develops theoretical foundations and inference methods for a class of nonlinear observation-driven time series models, demonstrating their properties and applying them to financial count data.

Contribution

It introduces a novel theoretical framework for nonlinear observation-driven models, establishing their Markov and mixing properties, and derives asymptotic inference results.

Findings

Conditional mean process is a geometric moment contracting Markov chain.

Observation process is absolutely regular with geometrically decaying coefficients.

Asymptotic normality of maximum likelihood estimates is established.

Abstract

This paper studies theory and inference related to a class of time series models that incorporates nonlinear dynamics. It is assumed that the observations follow a one-parameter exponential family of distributions given an accompanying process that evolves as a function of lagged observations. We employ an iterated random function approach and a special coupling technique to show that, under suitable conditions on the parameter space, the conditional mean process is a geometric moment contracting Markov chain and that the observation process is absolutely regular with geometrically decaying coefficients. Moreover the asymptotic theory of the maximum likelihood estimates of the parameters is established under some mild assumptions. These models are applied to two examples; the first is the number of transactions per minute of Ericsson stock and the second is related to return times of extreme events of Goldman Sachs Group stock.