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

Wagner Barreto-Souza, Ngai Hang Chan

TL;DR

The paper develops a generalised linear model framework for time series driven by latent processes, where $Y_t|\nu_t$ follows a bi-parameter exponential family with a multiplicative latent effect in the conditional mean. It proves the asymptotic normality of GLM estimators ignoring the latent process and derives a correct information matrix for valid inference, under strong mixing of the latent process. The authors also introduce prediction and forecasting methods based on $E(Y_{t+h}|Y_t)$, with explicit forms for log-normal AR(1), gamma AR(1) (GAR), and ARCH latent dynamics, and they illustrate the approach on measles infection data and paleoclimatic varves. The results demonstrate enhanced flexibility and improved predictive performance over standard GLMs, highlighting practical utility for counts and positive continuous time series in diverse domains.

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.

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

TL;DR

The paper develops a generalised linear model framework for time series driven by latent processes, where follows a bi-parameter exponential family with a multiplicative latent effect in the conditional mean. It proves the asymptotic normality of GLM estimators ignoring the latent process and derives a correct information matrix for valid inference, under strong mixing of the latent process. The authors also introduce prediction and forecasting methods based on , with explicit forms for log-normal AR(1), gamma AR(1) (GAR), and ARCH latent dynamics, and they illustrate the approach on measles infection data and paleoclimatic varves. The results demonstrate enhanced flexibility and improved predictive performance over standard GLMs, highlighting practical utility for counts and positive continuous time series in diverse domains.

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.
Paper Structure (8 sections, 2 theorems, 45 equations, 4 figures, 4 tables)

This paper contains 8 sections, 2 theorems, 45 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model specification
  3. Asymptotic distribution of the GLM estimators
  4. Prediction and forecasting
  5. Real data analyses
  6. Measles infections in North Rhine-Westphalia (Germany)
  7. Paleoclimatic glacial varves
  8. Concluding remarks

Key Result

Theorem 3.1

Assume that $\{\nu_t\}_{t\in\mathbb N}$ is a stochastic process satisfying (C1) and that the assumptions on covariates (as:mutmulcentre)-(as:sup) hold. Then, the following asymptotic normality is valid: as $n\rightarrow\infty$, where $\boldsymbol\beta_0$ denotes the true value of $\boldsymbol\beta$ and $\boldsymbol\Omega_{II}=\sum_{l=-\infty}^\infty\gamma_\nu(l){\bf W}_l$, with $\gamma_\nu(\cdot)

Figures (4)

  • Figure 1: Plots of the weekly reported cases of measles in North Rhine-Westphalia, Germany, between January 2001 and May 2013 and its respective autocorrelation function.
  • Figure 2: In-sample forecasting (dots) for the weekly reported cases of measles based on GLM, GLM-LNAR, GLM-GAR, and GLM-ARCH along with the observed time series (solid line).
  • Figure 3: Plots of the varve time series and its respective autocorrelation function.
  • Figure 4: In-sample forecasting (dots) for the varve time series based on GLM, GLM-LNAR, GLM-GAR, and GLM-ARCH along with the observed time series (solid line).

Theorems & Definitions (6)

  • Remark 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof