Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion

Abstract

The compound Poisson INAR(1) model for time series of overdispersed counts is considered. For such CPINAR(1) processes, explicit results are derived for joint moments, for the $k$-step-ahead distribution as well as for the stationary distribution. It is shown that a CPINAR(1) process is strongly mixing with exponentially decreasing weights. This result is utilized to design a test for overdispersion in INAR(1) processes and to derive its asymptotic power function. An application of our results to a real-data example and a study of the finite-sample performance of the test are presented.

Introduction

The study of count data time series (i.e., having a range contained in ${\mathbb{N}}_0=\{0,1,\dots \}$) has attracted a lot of attention in recent years. In particular, the integer-valued autoregressive model of order 1 (INAR(1) model) for stationary processes ${\left(Y_t\right)}_{t\in \mathbb{Z}}$ with an AR(1)-like dependence structure, as introduced by McKenzie (1985), was studied intensively, see Section  2.2 for details. This model has been applied to several real count data time series $y_1,\dots ,y_T$, see Freeland (1998), Freeland and McCabe (2004), Jung et al. (2005), Weiß (2008) as well as Section  5 below.

For such applications, especially the Poisson INAR(1) model, having a Poisson marginal distribution, would be an attractive choice for describing the data. One important characteristic of the Poisson distribution is equidispersion, i.e., its mean and variance are equal to each other. In contrast, many real-world data examples (see the above references) exhibit empirical overdispersion, i.e., the empirical variance is larger than the empirical mean. Typical reasons for observing overdispersion in practice are surveyed by Weiß (2009). But is the empirically observed degree of overdispersion really significant? And if so, how can we apply the INAR(1) model to overdispersed count data time series?

We first focus on the second question. Several modifications of the INAR(1) model have been suggested in the literature, usually changing the thinning operation used for defining the INAR(1) model (Weiß , 2008). The approach taken here focuses on the distribution of the innovations, by considering the larger class of compound Poisson $\left(\mathrm{CP}\right)$ distributions; a brief survey of the $\mathrm{CP}$-distribution is provided in Section  2. We introduce the compound Poisson INAR(1) model (CPINAR(1) model) which constitutes a convenient way to account for overdispersion in an INAR(1) process and comprises a number of specialized INAR(1) models within one model. To make the CPINAR(1) model broadly applicable, we derive its $k$-step-ahead forecast distribution, its stationary marginal distribution, closed-form expressions for its joint moments up to order 4 as well as mixing properties in Section  3.

We then turn our attention towards the first of the questions posed above and consider the (Poisson) index of dispersion, $I_Y$, which equals 1 in the case of the Poisson distribution. This index and its empirical counterpart are commonly defined as

$$I_Y≔\frac{{\sigma }_Y^2}{{\mu }_Y}\text{and}{\hat{I}}_Y≔\frac{S_Y^2}{\bar{Y}}\text{,}\text{ respectively.}$$

Here, $\bar{Y}=\frac{1}{T}{\sum }_{t=1}^T{Y}_t$ and $S_Y^2=\frac{1}{T}{\sum }_{t=1}^T{(Y_t-\bar{Y})}^2=\left(\frac{1}{T}{\sum }_{t=1}^T{Y}_t^2\right)-{\bar{Y}}^2$. Based on the results established for CPINAR(1) processes, especially concerning joint moments and mixing properties, we apply a central limit theorem, which, in turn, is used in Section  4 to derive closed-form expressions for the asymptotic distribution of the index of dispersion for CPINAR(1) processes. Utilizing these expressions, we develop a test based on the index of dispersion, and we investigate its power for uncovering overdispersion in INAR(1) processes. In Section  5, we apply our test to a real-data example. We conclude in Section  6 and outline issues for future research.