Estimation for Some Linear and Nonlinear Time Series Models 
Rongning Wu
Ph.D. Candidate, Department of Statistics, Colorado State University
Monday, June 18, 2007
9:30 a.m.
006 Statistics
ABSTRACT
This dissertation concerns parameter estimation for two different classes of models. One class is parameterdriven generalized linear models (GLMs) for time series, which is an important tool in modeling nonGaussian 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 pseudolikelihood. 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 parameterdriven GLMs for general time series, where the observation variable, conditional on covariates and a latent process, is assumed to have a distribution from a oneparameter 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 loglinear 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 autoregressivemoving 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 arameterization. In a nonGaussian 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 nonGaussian setups, and investigate exclusively least absolute deviation (LAD) estimation for general ARMA models, which is widely used in the nonGaussian setting, especially when observations are heavytailed. 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 deconstruct an ARMA(p,q) model into its causal, purely noncausal, invertible, and purely noninvertible components, and apply a local approximation technique to each component. We establish a functional limit theorem for random processes, from which the asymptotic results of the LAD estimator follow.
