Page 118 - Textos de Matemática Vol. 47

108 M. E. SILVA
In many cases, the discrete variates are large numbers and it may make sense to approximate them by continuous variates. Often, however, this is not possible or even desirable and it is necessary to develop an appropriate mod- elling strategy for the statistical analysis of time series of counts. One of the approaches developed is based on a random operation called thinning opera- tion capable of preserving the integer valued nature of the variables, giving rise to the class of INteger valued AutoRegressive, INAR, models. This paper considers the first order INAR, INAR(1), models for analyzing time series of counts and discusses associated statistical inference, comprising estimation, di- agnostics and model assessment. The plan of the paper is as follows. Section 2 introduces the INAR(1) models with including discussion of the most relevant properties. Section 3 considers parameter estimation and presents a set of tools appropriate to check the adequacy of fitted INAR models, an important part of any iterative modelling exercise in applied time series analysis. Section 4 illustrates the fitting of the models to a time series of counts of the stock type. Section 5 provides some concluding remarks.
2. First order INteger valued AutoRegressive models Definition 2.1. The first order integer autoregressive, INAR(1), model is de-
fined on the discrete support N0 by the recursive equation
Xt = ↵ ⇧ Xt 1 + ✏t, (2.1)
where {✏t} is a sequence of independent and identically distributed non-negative integer valued random variables, for each t independent of Xt 1 and of ↵⇧Xt 1, with finite mean μ✏ and variance ✏2 and, conditional on Xt 1, ↵ ⇧ Xt 1 is an integer valued random variable whose probability distribution depends on the parameter ↵.1
Thus, 0⇧02 denotes a random operator, usually called thinning operator, which always produces integer values and introduces serial dependence via the conditioning on Xt 1. Consider now that in model (2.1) we require that the marginal distribution of {Xt}t is of the same family as {✏t}. [25] proposes an approach to solve this problem within the convolution-closed infinitely divisible class of (marginal) distributions. The random operator 0⇧0 is required not only to introduce serial dependence and preserve the integer-valued status of the random variable but also to be unconditionally of the same family as {✏t}t.
1In fact, ↵ may be a vector of parameters but at this point we prefer the simpler scalar notation
2The operator is in fact ↵⇧ as it depends on ↵ but usually the simpler notation is used.