Estimation for some linear and nonlinear time series models

Abstract

This dissertation concerns parameter estimation for two different classes of models. One class is parameter-driven generalized linear models (GLMs) for time series, which is an important tool in modeling non-Gaussian time series. We first consider a negative binomial logit regression model for studying time series of count data. Serial dependence among observed data is introduced by incorporating a latent process in the link function of the model. We apply a standard GLM estimation by ignoring the latent process and maximizing the resulting pseudo-likelihood. We show the consistency and asymptotic normality of the GLM estimator under two cases: where the latent process is a stationary Gaussian process, and where it is a stationary strongly mixing process. We also study parameter-driven GLMs for general time series, where the observation variable, conditional on covariates and a latent process, is assumed to have a distribution from the one-parameter exponential family. We generalize the asymptotic results of the GLM estimator under suitable conditions, and thus unify in a common framework the results for Poisson log-linear regression models (Davis, Dunsmuir, and Wang, 2000), negative binomial logit regression models, and other models. Another class of models that we study consists of noncausal and/or noninvertible autoregressive-moving average (ARMA) models. The ARMA models are a class of linear time series models, which provides a general framework for studying stationary processes. In the classical Gaussian framework, causality and invertibility are assumed in order to eliminate the nonidentifiability of the parameterization. In a non-Gaussian setup, however, the assumptions are artificial because causal and noncausal (or invertible and noninvertible) models are identifiable. In this dissertation, we remove the assumption of causality and invertibility under non-Gaussian setups, and investigate exclusively least absolute deviation (LAD) estimation, which is widely used in the non-Gaussian setting, especially when observations are heavy-tailed. We first consider MA(1) models. Consistency and asymptotic normality are established for the local LAD estimator, the global LAD estimator, and the linearized LAD estimator in both the invertible and noninvertible cases. Then, we investigate LAD estimation for noncausal and/or noninvertible ARMA(p, q) models. We establish a functional limit theorem for random processes, from which the asymptotic results of the LAD estimator follow.