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Abstract

This paper introduces the PINAR(p)S model for periodic count time series with inflation of ze-

ros and covariates, denoted ZIP-PINAR(p)S model. The model properties such as stationarity

and ergodicity as well as the asymptotic properties of the conditional quasi-maximum likelihood parameter estimators are fully established. Simulations were carried out in order to verify the estimation method performance for finite sample sizes. The count time series of the number of visits to the hospital emergency service for people with respiratory diseases (asthma and rhinitis) was analyzed using the proposed model with the covariate air pollutant concentra- tions.

Keywords: INAR, periodic stationarity, PINAR, zero inflated Poisson, quasi-maximum likeli- hood.

1

Introduction

In the literature of time series, a series with a excessive number of zeros is usually defined as time series with zero-inflation and this phenomenon is quite common in many area of applica- tion. For example, in the biomedical and public health domains, some types of rare diseases with low infection rates can lead to a count time series with a large number of zeros. Ignoring zeros in the data may conduct to a wrong model choice and inference and a spurious associa- tion between the count time series with covariates.

Yang et al. (2013) extended the classical regression based on the Zero Inflated Poisson (ZIP) distribution, introduced by Lambert (1992), for counting time series with an excess of zeros, autoregressive (AR) structure and time-dependent covariates regression framework. The ZIP distribution can be seen as a mixed distribution of a Poisson with parameter λ, and a degenerate component with all its mass at zero, parameter ρ, called the inflation parameter of zeros. The regression ZIP is referred here as ZIP model.

To illustrate the mechanism of this model, consider the process {εt}t∈Z of Z+-valued indepen-

dent ZIP distributed random variables (r.v.’s) εt, with inflation of zeros parameter ρ ∈ [0; 1] and

Poisson parameter λ ∈ R+, εt∼ ZIP(ρ, λ). The probability mass function (p.m.f.) of εt is given

by Pεt(εt = m) = ρIm=0 + (1− ρ) exp(−λ)λ

m_{/m!, m ∈ Z}

+, where Im=0 = 1, if m = 0 or

Im=0 = 0, if m 6= 0. The parameters ρ and λ relate the variable ε to the vectors of covariates X

and Z through equations log(λ) = X>_{β and log[ρ/(1 − ρ)] = Z}>_{γ, where β and γ represent}

A simple model for a stationary sequence of integer-valued random variables with lag-one dependence referred to as the integer-valued autoregressive of order one 1 (INAR1) model was introduced by Al-Osh & Alzaid (1987). This model has a special advantage over the ZIP model due its similarity to the Box and Jenkins ARMA models for continuous data. INAR model has the same additive structure of ARMA models instead of the multiplicative structure presented in ZIP. This additive characteristic and the discreteness of the modeled process are proportioned by the Thinning Operator ◦. INAR model can be represented as the process Yt= α◦ Yt−1+ εt,

where α ◦ Yt−1 = Y_Xt−1

i=1

Bi(α). Bi(α) represents a sequence of independent random variables

(R.V.s) with Bernoulli distribution and probability of success P(Bi,t(α) = 1) = α, 0 ≤ α ≤ 1. In

this case ◦ is called binomial thinning operator. For the Poisson INAR model, εt∼ P oisson(λ)

represents a sequence of independent R.V.s, assumed independent of Yt−1and α ◦Yt−1.

A natural extension of Poisson INAR model to provide a consistent fit to count times series with over-dispersion proportioned by the excess of zeros is the first-order integer valued AR pro- cesses with zero inflated Poisson innovations (ZINAR) developed by Jazi et al. (2012). The ZI- NAR model has similar equation of INAR models, in which the innovations εt∼ ZIP(ρ, λ).

The inclusion of explanatory variables to extend the applicability of INAR models was briefly introduced by Br¨ann¨as (1993) and studied by Enciso-Mora et al. (2009). The former authors introduced a model based on the INAR(p) model and developed an efficient Markov Chain Monte Carlo algorithm which analyze both explanatory variable an model order selection. The methodology was applied to the analysis of monthly polio incidences in the USA 1970-1983 and claims from the logging industry to the British Columbia Workers’ Compensation Board 1985-1994.

The models discussed in the previous paragraphs are based on the assumption of stationarity in the mean and variance, that is, standard count time series models. However, it is quite common in many area of application to have time series that varies periodically in the mean, the variance and the autocovariance. This type of time series was introduced by Gladyshev (1961) as Periodically Correlated process (PC).

There is a lot of research related to PC processes for continuous time series. For a review, from theoretical and applied point of view, of the periodic autoregressive moving-average (PARMA) models, one can mention Gardner et al. (2006), Sarnaglia et al. (2010), Basawa & Lund (2001) among others. However, no much attention has been paid to the analysis of periodically corre- lated count series. For example, Monteiro et al. (2010) introduced the periodic integer-valued autoregressive model of order 1, with Poisson distributed data (PINAR). The stationarity and ergodicity properties of the process were established following the same lines in Latour (1997). The Yule-Walker based equations, least squares-type and quasi-maximum likelihood estima- tors of the model parameters were the estimation methods discussed.

Filho et al. (n.d.) extended the PINAR(1)Smodel to the PINAR(p)S. Statistical properties of the

model such as mean, variance, marginal and joint distributions are presented in the paper. The Moments-based, conditional least squares and quasi-maximum likelihood estimation methods

of the parameters were considered. An application to medication dispensing is given to show the usefulness of the proposed model.

Here, it is introduced an extension of PINAR(p)S model, denoted as ZIP-PINAR(p)S model,

which takes into account periodic cont time series with zero-inflates and covariates as explana- tory variables of the model.

In the remainder of this paper, let N, Z, Z+, R and R+ denote the set of positive integers,

integers, non-negative integers, real numbers, and non-negative real numbers, respectively. Denote by Idthe d × d identity matrix. If it is clear from the context, then we omit the subscript

d. Bin(n, α) denotes a binomial distribution with parameters n ∈ N and α ∈ [0, 1]; Poi(λ) denotes a Poisson distribution with mean parameter λ ∈ R+. Let E(·) and E(·|·) represent the

expectation and the conditional expectation, respectively. Random variables are all defined on a common probability space (ω, A, P) and Ft denotes the σ-algebra generated by the random

variables until time t, for all t ∈ Z, t > S and F0 ={∅, ω}.

The organization of the paper is as follows: Section 2 presents the PINAR(p)S model; Section

3 introduces the regression ZIP-PINAR(p)S model, and some of its statistical and probabilistic

properties; The transition probability function of the process established on the ZIP model is introduced in Section 4; the Section 5 discusses the Quasi-Maximum Likelihood(QML) estima- tion method of the parameters of the model; Section 6 presents a set of simulations; a real data application is the motivation of the Section 7, and, finally, conclusions and final comments are presented at the last section. Some proofs are in the Appendix.

2

The PINAR(p)

model

Let {Yt}t∈Z, Yt∈ Z+, be a stochastic process with seasonal characteristics of period S, S ∈ N.

The time index t may be written as the Euclidean division between t and S, i.e., as t = kS + ν, where k ∈ Z and ν = 1, . . . , S. For example, in the case of daily data, S = 7, ν and k represent the day of the week and the week, respectively.

Define the mean function, µ(t) = E(Yt) for all t ∈ Z, and the covariance function, the scalar

γk,ν, ν = 1, . . . , S and k ∈ Z, on Z as

γk,ν(h) = Cov(YkS+ν, YkS+ν−h), h∈ Z. (5.1)

The stochastic process {YkS+ν}k∈Z,ν=1,...,S, satisfying E(YkS+ν2 ) <∞, is said to be a periodically

correlated process(PC) of period S, S ∈ N, if, for ν = 1, . . . , S and all integers k, µ(kS + ν) = µν and γk,ν(h) = γν(h).

That is, if mean and variance are finite and if they do not depend on k. Note that, if {Yt}t∈Z

is a periodically correlated process then its mean and covariance are periodic functions with period S. If S = 1, then {Yt} represents a homogeneous stochastic process and the condition

of periodic stationarity is equivalent to the non-periodic ones.

A Z+-valued process {Yt} is said to be a periodic non-negative integer-valued autoregression

(PINAR) with seasonal period S, for some S ∈ {2, 3, . . .}, and is denoted by PINAR(p)S, where

p =max(~p) and ~p is the 1 × S vector of autoregressive orders of {Yt}, if it satisfies the following

stochastic recursion Yν+kS= pν X i=1 αi(ν)◦ Yν−i+kS+ εν+kS, (5.2)

where k ∈ Z and t = ν +kS. In this paper assume that p ≤ S. Because of the similarity of INAR models to the standard autoregressive (AR) model for continuous data, the αi(ν), i = 1, . . . , pν,

ν = 1, 2, . . . , S and pν = 1, . . . , S, are called autoregressive coefficients. The vector of AR

orders ~p has the form (p1, p2, . . . , pS)1×S, S ∈ N , where pν represents the AR order of the ν−th

season. For each season ν = 1, 2, . . . , S, the set of autoregressive coefficients has the form {α1(ν), . . . , αpν(ν)} ⊂ [0; 1]

pν. The immigration process {ε

t}t∈Z is a periodic sequence of Z+-

valued r.v.’s such that for each ν ∈ {1, . . . , S} the sequence {εkS+ν}k∈Z consists of i.i.d.r.v.’s.

The r.v.’s Y1, . . . , YSare known as the starting values for the recursion (5.2). Finally, in (5.2), we

assume that all counting random variables are mutually independent and they are independent of the sequence {εt}t∈Z.

It is assumed that the immigration process and starting values have finite second moments, i.e., the mean function µ and the variance of {Yt}t∈Z exist and are finite. Moreover, let µε(ν) =

E(εkS+ν), σ2ν = Var(εkS+ν)for all k ∈ Z and ν = 1, . . . , S. As can be seen that in the seasonal

period ν, Yt in (5.2) has pν + 1 random components; the immigration part of the past Yt−i,

t = ν + kS and i = 1, . . . , pν, with survival probability αi(ν)and the elements which entered

in the system in the interval (t-1, t], which define the innovation term εt for all t ∈ Z. More-

over, the autoregressive parameters αi(ν)and immigration means µε(ν), ν = 1. . . . , S, change

periodically according to the seasonal period S.

The mean of the process {Yt}t∈Z, t > p, where p = max(~p), in (5.2) is given by:

µ(kS + ν) =

pν

X

i=1

αi(ν)µ(kS + ν− i) + µε(ν). (5.3)

A PINAR(p)S model is called proper if its parameters αi(ν), µε(ν), 1 ≤ i ≤ p and ν = 1, . . . , S

are positive values. Consider that {Yt}t∈Z is a proper PINAR(p)Smodel and satisfies all of the

statements of Lemma 1 in PINAR(p)S paper. Then, from the Theorem 1 in PINAR(p)S paper,

the process {Yt}t∈Z is a second order periodically stationary process.