Multivariate Count Autoregressive Models and their Assessment

Count time series data are found in diverse applications arising in the field of Economic and Social Statistics. For example, the Office of National Statistics (ONS), publish time series data sets on business activity and demographics, labour market status, people and population, and so on. Much of these data are multivariate integer-valued time series that consist of counts and there is considerable lack of methodology for their proper analysis. Moreover, new data sources to integrate with or replace traditional surveys are being explored by ONS and the wider Government Statistical Service to improve understanding of the UK's economy, society and population, with time series analysis playing an important role. For analysing integer-valued time series, it is usually hard to impose a full parametric model for developing inference, as there are many available versions of the multivariate Poisson distribution in the literature. Even if one identifies a suitable multivariate model, the computations involved are usually cumbersome. In addition, a parametric model might
Last Updated August 2018
miss the true correlation form that might exist in the data (consider, for example, number of car accidents or number of unemployed people by neighbourhing regions) and not be able to describe marginally the phenomenon of overdispersion that is usually found in count data. These points, and others, have been thoroughly discussed in the recent work by Fokianos et al (2018) who provides a framework for building observation-driven autoregressive linear and log-linear models for multivariate count time series. The point of view of these authors is based on generalized linear models' methodology as advocated by McCullagh and Nelder (1989). Fokianos et al (2018) have suggested a data generating process that does not necessarily impose marginally a Poisson assumption, yet the structure of the proposed models is kept simple. Estimation of unknown matrix parameters is implemented by Quasi Maximum Likelihood Estimation (QMLE, see Heyde (1997)) and, under mild conditions, it is shown that these estimators possess good properties. This work will be the basis for developing further methodology for multivariate count autoregressions motivated by real ONS data. The combination of likelihood inference and generalized linear models provide a systematic framework for the analysis of quantitative as well as qualitative time series data. Indeed, estimation, diagnostics, model assessment, and forecasting are implemented in a straightforward manner where computations can be easily developed.