Parameter Change Test for Time Series of Counts

Abstract

In this thesis, we consider parameter change test for time series of counts. First we consider the problem of testing for parameter change in zero-inflated generalized Poisson (ZIGP) autoregressive models. We verify that the ZIGP process is stationary and ergodic and that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal. Then, based on these results, we construct CMLE- and residual-based cumulative sum tests and show that their limiting null distributions are a function of independent Brownian bridges. Simulation results are provided for illustration and a real data analysis is performed on data of crimes in Australia. Second we consider bivariate Poisson integer-valued GARCH(1,1) models, constructed via a trivariate reduction method of independent Poisson variables. We verify that the CMLE of the model parameters is asymptotically normal. Then, based on these results, we construct CMLE- and residual-based CUSUM tests and derive that their limiting null distributions are a function of independent Brownian bridges. A simulation study are conducted for illustration. We analyze two daily data sets of car accidents that occurred in Sungdong and Seocho counties in Seoul, Korea. Finally, we consider the problem of testing for a parameter change in general nonlinear integer-valued time series models where the conditional distribution of current observations is assumed to follow a one-parameter exponential family. We consider score-, (standardized) residual-, and estimate-based CUSUM tests, and show that their limiting null distributions take the form of the functions of Brownian bridges. Based on the obtained results, we then conduct a comparison study of the performance of CUSUM tests, through the use of Monte Carlo simulations. Our findings demonstrate that the standardized residual-based CUSUM test largely outperforms the others.