Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

Abstract

In this article, we propose a first‐order integer‐valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero‐modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero‐modified geometric INAR(1) process. This article contains a Supporting Information.

On two classes of reflected autoregressive processes

“…Such an extension of the INAR(1) process was proposed by [17]. Reference [21] discusses the case that the increments corresponding to R n ( • ) have a geometric distribution; see also [4].…”

A zero-inflated geometric INAR(1) process with random coefficient

“…Random coefficient INAR(1) processes, in which the autoregressive parameter itself is a random variable, are considered by a few authors such as [9], [27], [25]. Further, [11], [14] introduced INAR(1) models with zero-inflated Poisson and zero-inflated generalized power series innovations, respectively, and [7] introduced a zero-modified geometric INAR(1) model based on negative binomial thinning operator for modeling count series with inflation or deflation of zeros. Recently, [21] introduced a general family of INAR(1) models with compound Poisson innovations.…”

“…Therefore, this approach is different from the one used in [11], [14], as the latter specify the INAR(1) model by assuming a certain distribution for the innovations. Further, our approach defines an INAR(1) model with a random coefficient, which is different from the model of [7] because of the nature of the autoregressive parameter of the INAR(1) model, α. To show this let X t be the number of patients in inpatient wards in the t-th month, hence X t obeys an INAR process and represents the sum of the number of surviving patients from the previous month (denoted by α • X t−1 ) and the newly admitted patients in the current month (ε t ).…”

“…For selecting the best model for the data series among a finite set of INAR(1) models, we compared the ZIGINAR RC (1) model to some competitive INAR(1) models with geometric and negative binomial marginals which are GINAR(1) ( [4]), NBI-INAR(1) ( [1]), NBRCINAR(1) ( [24]), NGINAR(1) ( [19]), ZMGINAR(1) ( [7]) and GINAR RC (1) (Remark 2.1). For each INAR model, we obtained the maximum likelihood estimates, the four goodness-of-fit statistics (AIC, BIC, HQIC and CAIC) and the root mean squares of the differences of observations and predicted values (RMS).…”

Modeling continuous time series with many zeros and an application to earthquakes

“…Zhu () proposed zero‐inflated Poisson and negative binomial integer‐valued generalized autoregressive conditional heteroskedastic (GARCH) models, which are a mixture of ZIP/zero‐inflated negative binomial and integer‐valued GARCH models (Ferland, Latour, & Oraichi ), to model integer‐valued time series with zero inflation and overdispersion. Other works that recently appeared in the literature dealing with zero inflation in integer‐valued time series include Zhu (), Yang, Zamba, and Cavanaugh (), Barreto‐Souza (), Li, Wang, and Zhang (), Yang, Cavanaugh, and Zamba (), Chen and Lee (), Gonçalves, Mendes‐Lopes, and Silva (), and Rakitzis, Weiß and Castagliola ().…”