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

where ν t is a log-normal AR(1) process with mean 1 and some finite variance, x t is a vector of covariates, and β is the associated vector of regression coefficients. Among the asymptotic properties explored by the authors, they established conditions to ensure asymptotic normality of the GLM estimators based on the Poisson likelihood by ignoring the latent process. The model (1) was also previously studied by Chan & Ledolter (1995), where a Monte Carlo EM algorithm was developed for estimation of the parameters. A two-step estimation procedure with the regression parameters being estimated from the marginal distribution, and the parameters of the latent process estimated via a composite likelihood approach, was proposed by Sørensen (2019). Davis & Wu (2009) extended model ( 1) by replacing the Poisson assumption by the oneparameter exponential family (EF), with focus on the negative binomial case (with known dispersion parameter). The extended model assumes that Y t given a stationary strongly mixing latent process ν t follows an one-parameter exponential family with conditional mean

where h(•) is the inverse of a link function such that the resulting GLM likelihood is concave and E(h(x ⊤ t β + ν t )) = h(x ⊤ t β). Under the above formulation, such a function exists, for instance, for Poisson, negative binomial, and Gaussian cases. The asymptotic normality of the GLM estimators was established based on the GLM likelihood by ignoring the latent process.

Other related recent contributions include Maia et al. (2021), where a class of semiparametric time series was proposed by using quasi-likelihood models driven by latent processes, and Barreto-Souza & Ombao (2022), where a Poisson regression driven by a gamma AR(1) process was developed with a composite likelihood inferential approach.

Our chief goal in this paper is to introduce a flexible GLM driven by latent processes to handle counts, real-valued, continuous, positive continuous time series. To do this, we assume that the time series of interest, say {Y t }, conditional on a latent process {ν t }, follows a bi-parameter exponential family, with a conditional mean having the latent process in a multiplicative way, and with a dispersion parameter, which can be unknown. We establish the asymptotic normality of the GLM estimators and provide the the correct information matrix to assess standard errors of the parameter estimates. Our formulation have some advantages when compared to the Davis & Wu (2009)’s model as follows: (i) by entering the latent effect in a multiplicative way instead of additive like in (2), it allows us to handle a gamma time series to address positive continuous time series, which is not feasible under the existing approach (more details on that are provided in Remark 2.1); (ii) we consider a dispersion parameter that can be estimated based on the method of moments, so adding more flexibility; (iii) if the function h(•) is not exponential, then a multiplicative latent effect form is not possible and therefore it is hard to ensure the requirements by Davis & Wu (2009) to establish the asymptotics for the GLM estimators such as E(h(x ⊤ t β + ν t )) = h(x ⊤ t β) and their condition stated in Theorem 3 (weak convergence of their C n (s) quantity); under our formulation, these issues are easily addressed; (iv) we also address how to predict based on GLMs driven by latent processes, which is not addressed in the current literature; (v) we consider different latent processes in our numerical results, while a log-normal AR(1) assumption has been usually adopted in the literature; for example, our empirical illustrations show that a gamma AR(1) latent process might provide better results when compared to the log-normal AR(1) process.

This paper is organised as follows. Section 2 introduces our class of GLMs driven by latent processes and provide some basic results. Section 3 establishes the asymptotic normality of the GLM estimators based on the Central Limit Theorem for strongly mixing processes by Peligrad and Utev (1997), and explicitly provide the correct information matrix to assess the standard errors for valid inference. Prediction and forecasting are addressed in Section 4. Section 5 presents two real data applications on measles infection cases in North Rhine-Westphalia (Germany) and paleoclimatic glacial varves time series, which illustrates the Poisson and gamma GLM time series models’ performance in practice. Concluding remarks are presented in Section 6.

We start this section by introducing key ingredient