On variance estimation in a negative binomial time series regression model

We study variance estimation in a negative binomial regression model for analyzing time series of counts, where serial dependence among the observed counts is introduced by an autocorrelated latent process. The regression coefficient vector is estimated by maximizing the pseudo-likelihood with the latent process suppressed. The resulting estimator is referred to as the generalized linear model estimator, and its consistency and asymptotic normality have been established by Davis and Wu [R.A. Davis, R. Wu, A negative binomial model for time series of counts, Biometrika 96 (2009) 735–749] when the latent process is stationary and strongly mixing. However, in order to perform valid statistical inferences about the regression coefficients, it is essential to develop a consistent estimation procedure for the asymptotic covariance matrix of the generalized linear model estimator. We propose two types of estimators using kernel-based and subsampling methods, and establish their consistency property. The results can be generalized straightforwardly to time series following a parameter-driven generalized linear model. Simulation study is conducted to evaluate the finite sample performance of the estimation methods.