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Count time series, Parameter-driven, Observation-driven, Autocorrelation, Overdispersion, Diagnostics,

The aim of the research that prompted this series of papers was to provide the Office of the Deputy Prime Minister (ODPM) with a migration modelling system, MIGMOD, potentially capable of informing the national government about policies that might have an impact on population redistribution within England and Wales. Some current concerns, for example, include: the drift of population to the South East, with concomitant increases in land pressure and house prices in that region; long-standing regional problems that are compounded by the loss of skilled population from economically depressed areas; and the trend of counterurbanisation involving a net loss of internal migrants from major urban areas to surrounding rural areas.

In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first-order and second-order autoregressive as well as first-order moving average processes.

Bias correction, Estimation, INAR models, Overdispersion, Small sample properties, Time series of counts,

This article considers some simple observation-driven time series models for counts. We provide a brief description of the class of integer-valued autoregressive (INAR) and integer-valued moving average (INMA) processes. These classes of models may be attractive when the data exhibit a significant serial dependence structure. We, therefore, briefly review various testing procedures useful for assessing the serial correlation in the data. Once it is established that the data are not serially independent, suitable INAR or INMA processes may be employed to model the data. In the important first order INAR model, we discuss various methods of estimating the structural parameters of the process. We also give a short account of the extension of some of these estimation procedures to second order INAR models. Moving average counterparts of both models are also entertained. Throughout, the models and methods are illustrated in the context of a famous data set from the branching process literature that turns out to be surprisingly difficult to model satisfactorily.