Generalised Linear Models Driven by Latent Processes: Asymptotic Theory and Applications

TL;DR

This paper develops a broad class of latent-process driven generalized linear models that extend existing frameworks, allowing for more flexible distributions, improved inference, and applications to real-world count, binary, and continuous data.

Contribution

It introduces a generalized latent-process GLM framework with bi-parameter exponential family responses, asymptotic theory, and prediction methods, broadening the scope of previous models.

Findings

Asymptotic normality of estimators established

Method for estimating dispersion parameters introduced

Applications demonstrate model flexibility and practical advantages

Abstract

This paper introduces a class of generalised linear models (GLMs) driven by latent processes for modelling count, real-valued, binary, and positive continuous time series. Extending earlier latent-process regression frameworks based on Poisson or one-parameter exponential family assumptions, we allow the conditional distribution of the response to belong to a bi-parameter exponential family, with the latent process entering the conditional mean multiplicatively. This formulation substantially broadens the scope of latent-process GLMs, for instance, it naturally accommodates gamma responses for positive continuous data, enables estimation of an unknown dispersion parameter via method of moments, and avoids restrictive conditions on link functions that arise under existing formulations. We establish the asymptotic normality of the GLM estimators obtained from the GLM likelihood that ignores the latent process, and we derive the correct information matrix for valid inference. In addition, we provide a principled approach to prediction and forecasting in GLMs driven by latent processes, a topic not previously addressed in the literature. We present two real data applications on measles infections in North Rhine-Westphalia (Germany) and paleoclimatic glacial varves, which highlight the practical advantages and enhanced flexibility of the proposed modelling framework.