Modeling time series of count data

Abstract

The focus of this thesis is on modeling time series of count data. We consider an extension of linear Gaussian state space models - parameter driven models in which th e mean function of a time series of observed counts {Yt} is specified bv a linear predictor modified by a 'latent process’. As in linear regression with correlated errors, there is a need for model diagnostic and identification techniques to decide if it is necessary to include a latent process in the specification of the mean of the Poisson counts and, if so, is there any evidence of autocorrelation in such a process.

For a parameter driven model, the asymptotic distribution of standard generalized linear model estimators is derived for the case that an autocorrelated strong mixing latent process is present. Simple formulas for the effect of the autocovariance of the latent process on standard errors of the regression coefficients are also provided. A method of testing for the existence of a latent process is developed and compared to existing test statistics via simulation. Once the existence of a latent process has been detected, a simple and easily implementable method for estimating the autocovariance of the latent process is given. The standard errors of the estimates are also provided. Methods for adjusting for severe bias in previously proposed estimators of autocovariance are derived and their behavior investigated. A test statistic for testing serial dependence in the latent process is proposed based on the study of the distribution of autocorrelation estimates. The performance of different test statistics for testing serial dependence have been compared.

Parameter estimation of a parameter driven model is complicated since the latent process is unobservable and the likelihood of the observed data y is an n-fold integral which does not have a simple closed form. Existing estimation methods involve intensive Monte Carlo simulation. A new estimation method that avoids Monte Carlo simulation is developed using an approximation to the likelihood of y. Applications of the methods to time series of monthly polio counts in the U.S . is used to illustrate the methods and results. A simulation study has been conducted to compare the performance of the various estimation methods.