Abstract
The study of time series models for count data has become a topic of special interest during the last years. However, while research on univariate time series for counts now flourishes, the literature on multivariate time series models for count data is notably more limited. The main reason for this is that the analysis of multivariate counting processes presents many more difficulties. Specifically, the need to account for both serial and cross-correlation complicates model specification, estimation and inference. This thesis deals with the class of INteger-valued AutoRegressive (INAR) processes, a recently popular class of models for time series of counts. The simple, univariate INAR(1) process is initially extended to the 2-dimensional space. In this way, a bivariate (BINAR(1)) process is introduced. Subsequently, the time invariant BINAR(1) model is generalized to a BINAR(1) regression model. Emphasis is given on models with bivariate Poisson and bivariate negative binomial innovat ...
The study of time series models for count data has become a topic of special interest during the last years. However, while research on univariate time series for counts now flourishes, the literature on multivariate time series models for count data is notably more limited. The main reason for this is that the analysis of multivariate counting processes presents many more difficulties. Specifically, the need to account for both serial and cross-correlation complicates model specification, estimation and inference. This thesis deals with the class of INteger-valued AutoRegressive (INAR) processes, a recently popular class of models for time series of counts. The simple, univariate INAR(1) process is initially extended to the 2-dimensional space. In this way, a bivariate (BINAR(1)) process is introduced. Subsequently, the time invariant BINAR(1) model is generalized to a BINAR(1) regression model. Emphasis is given on models with bivariate Poisson and bivariate negative binomial innovations. The properties of the BINAR(1) model are studied and the methods of moments, Yule-Walker and conditional maximum likelihood are proposed for the estimation of its unknown parameters. Issues of diagnostics and forecasting are considered and predictions are produced by means of the conditional forecast distribution. A generalized specification of the BINAR(1) process, where cross-correlation between the two series receives contribution from two different sources, is also discussed. We mainly focus on the assumption of bivariate Poisson innovations. The resulting joint distribution of the bivariate series is identified as an 8-parameters bivariate Hermite. At a second stage, the BINAR(1) process is extended to the multi-dimensional space. Thus, we define a multivariate integer-valued autoregressive process of order 1 (MINAR(1)) and examine its basic statistical properties. Such an extension is not simple and we emphasize on problems that occur relating to selecting a reasonable innovation distribution as well as on problems related to inference. We also study two specific parametric cases that arise under the assumptions of a multivariate Poisson and a multivariate negative binomial distribution for the innovations. To overcome the computational difficulties of the maximum likelihood approach we suggest the method of composite likelihood. Extensions to incorporate covariance information are also discussed. The proposed models are illustrated on real data series.
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