Nonstationary INAR(1) Process with 𝑞th-Order Autocorrelation Innovation

Abstract

This paper is concerned with an integer-valued random walk process with qth-order autocorrelation. Some limit distributions of sums about the nonstationary process are obtained. The limit distribution of conditional least squares estimators of the autoregressive coefficient in an auxiliary regression process is derived. The performance of the autoregressive coefficient estimators is assessed through the Monte Carlo simulations.

1. Introduction

In many practical settings, one often encounters integer-valued time series, that is, counts of events or objects at consecutive points in time. Typical examples include daily large transactions of Ericsson B, monthly number of ongoing strikes in a particular industry, number of patients treated each day in an emergency department, and daily counts of new swine flu cases in Mexico. Since most traditional representations of dependence become either impossible or impractical; see Silva et al. [1], this area of research did not attract much attention until the early 1980s. During the last three decades, a number of time series models have been developed for discrete-valued data. It has become increasingly important to gain a better understanding of the probabilistic properties and to develop new statistical techniques for integer-valued time series.

The existing models can be broadly classified into two types: thinning operator models and regression models. Recently the thinning operator models have been greatly developed; see a survey by Weiß [2]. A number of innovations have been made to model various integer-valued time series. For instance, Ferland et al. [3] proposed an integer-valued GARCH model to study overdispersed counts, and Fokianos and Fried [4], Weiß [5], and Zhu and Wang [6–8] made further studies. Bu and McCabe [9] considered the lag-order selection for a class of integer autoregressive models, while Enciso-Mora et al. [10] discussed model selection for general integer-valued autoregressive moving-average processes. Silva et al. [1] addressed a forecasting problem in an INAR(1) model. McCabe et al. [11] derived efficient probabilistic forecasts of integer-valued random variables. Several other models and thinning operators were proposed as well.

Random coefficient INAR models were studied by Zheng et al. [12, 13] and Gomes and e Castro [14], while random coefficient INMA models were proposed by Yu et al. [15, 16]. Kachour and Yao [17] introduced a class of autoregressive models for integer-valued time series using the rounding operator. Kim and Park [18] proposed an extension of integer-valued autoregressive INAR models by using a signed version of the thinning operator. The signed thinning operator was developed by Zhang et al. [19] and Kachour and Truquet [20].

However, these models focus exclusively on stationary processes, whereas nonstationary processes are often encountered in reality. Some studies have examined asymptotic properties of nonstationary INAR models for nonstatinaory time series. Hellström [21] focused on the testing of unit root in INAR(1) models and provided small sample distributions for the Dickey-Fuller test statistic. Ispány et al. [22] considered a nearly nonstationary INAR(1) process. It was shown that the limiting distribution of the conditional least squares estimator for the coefficient is normal. Györfi et al. [23] proposed a nonstationary inhomogeneous INAR(1) process, where the autoregressive type coefficient slowly converges to one. Kim and Park [18] considered a process called integer-valued autoregressive process with signed binomial thinning to handle a nonstationary integer-valued time series with a large dispersion. Drost et al. [24] studied the asymptotic properties of this “near unit root” situation. They found that the limit experiment is Poissonian. Barczy et al. [25] proved that the sequence of appropriately scaled random step functions formed from an unstable INAR(p) process that converges weakly towards a squared Bessel process.

However, to our knowledge, few effort has been devoted to studying nonstationary INAR(1) models with an INMA innovation. In this paper, we aim to fill in this gap in the literature. We consider a new nonstatioary INAR(1) process in which the innovation follows a qth-order moving average process (NSINARMA(1,𝑞)). Similar to the studies of nonstationary processes for continuous time series models, we need to accommodate two stochastic processes, one as the “true process” and the other as an “auxiliary regression process.” In this paper, we study particularly statistical properties of the conditional least squares (CLS) estimators of the autoregressive coefficient in the auxiliary integer-valued autoregression process, when the “true process” is nonstationary integer-valued autoregression with an innovation which has an integer-valued moving average part.

The rest of this paper is organized as follows. In Section 2, our nonstationary thinning model with INMA(𝑞) innovation is described, and some statistical properties are established. In Section 3, the limiting distribution of the autoregressive coefficient in the auxiliary regression process is derived. In Section 4, simulation results for the CLS estimator are presented. Finally, concluding remarks are made in Section 5.

2. Definition and Properties of the NSINARMA(1,𝑞) Process

In this section, we consider a nonstationary INAR(1) process which can be used to deal with the autocorrelation of innovation process.

Definition 1. An integer-valued stochastic process {𝑋_𝑡} is said to be the NSINARMA(1,𝑞) process if it satisfies the following recursive equations: 𝑋_𝑡=𝑋_𝑡−1+𝑢_𝑡,𝑢_𝑡=𝜀_𝑡+𝜃₁∘𝜀_𝑡−1+𝜃₂∘𝜀_𝑡−2+⋯+𝜃_𝑞∘𝜀_𝑡−𝑞, (1) where {𝜀_𝑡} is a sequence of i.i.d. nonnegative integer-valued random variables with finite mean 𝜇_𝜀<∞, variance 𝜎²_𝜀<∞, and probability mass function 𝑓_𝜀. All counting series 𝜃_𝑘∘𝜀_𝑡−𝑘 are mutually independent, and 𝜃_𝑘∈[0,1],  𝑘=1,…,𝑞. For the sake of convenience, we suppose that the process starts from zero, more precisely, 𝑋₀=0.

It is easy to see the following results of 𝑋_𝑡 hold.

Proposition 2. For 𝑡≥1, one has(i)𝐸(𝑋_𝑡𝑋_𝑡−1)=𝑋_𝑡−1+(1+𝜃₁+⋯+𝜃_𝑞)𝜇_𝜀,(ii)𝐸(𝑋_𝑡)=𝑡(1+𝜃₁+⋯+𝜃_𝑞)𝜇_𝜀,(iii)Var(𝑋_𝑡𝑋_𝑡−1)=𝜇_𝜀∑^𝑞_𝑘=1𝜃_𝑘(1−𝜃_𝑘)+𝜎²_𝜀(1+∑^𝑞_𝑘=1𝜃²_𝑘),(iv)Var(𝑋_𝑡)=𝑡(𝜇_𝜀∑^𝑞_𝑘=1𝜃_𝑘(1−𝜃_𝑘)+𝜎²_𝜀(1+∑^𝑞_𝑘=1𝜃²_𝑘)).

Proof. It is straightforward to get (i) to (iii). We prove (iv) by induction with Var𝑋_𝑡=Var𝐸𝑋_𝑡∣𝑋_𝑡−1+𝐸Var𝑋_𝑡∣𝑋_𝑡−1=Var𝑋_𝑡−1+𝜇_𝜀^𝑞_𝑘=1𝜃_𝑘1−𝜃_𝑘+𝜎²_𝜀1+^𝑞_𝑘=1𝜃²_𝑘 (2) and the initial value 𝑋₀=0.

3. Estimation Methods

Similar to the continuous time series process with unit roots, we focus on the properties of the autoregressive coefficient estimator in an auxiliary regression process when the true process follows the nonstatioary INAR(1) process defined as Definition 1.

Suppose that the auxiliary regression process 𝑋_𝑡 is an INAR(1) model, 𝑋_𝑡=𝛼∘𝑋_𝑡−1+𝑣_𝑡, (3) where {𝑣_𝑡} is a sequence of i.i.d. nonnegative integer-valued random variables with finite mean 𝜇_𝑣 and variance 𝜎²_𝑣. We are interested in the properties of 𝛼 when the true process is a nonstatioary INAR(1) model with moving average components. In this paper, we consider a conditional least squares (CLSs) estimator. An advantage of this method is that it does not require specifying the exact family of distributions for the innovations.

Let 𝑄𝛽=^𝑇_𝑡=1𝑋_𝑡−𝛼∘𝑋_𝑡−1−𝜇_𝑣², (4) with 𝛽=(𝛼,𝜇_𝑣), be the CLS criterion function. The CLS estimators of 𝛼 and 𝜇_𝑣 are obtained by minimizing 𝑄 and are given by 𝛼=𝑇∑^𝑇_𝑡=1𝑋_𝑡−1𝑋_𝑡−∑^𝑇_𝑡=1𝑋_𝑡−1∑^𝑇_𝑡=1𝑋_𝑡𝑇∑^𝑇_𝑡=1𝑋²_𝑡−1−∑^𝑇_𝑡=1𝑋_𝑡−1²,𝜇_𝑣=𝑇^−1𝑇_𝑡=1𝑋_𝑡−𝛼^𝑇_𝑡=1𝑋_𝑡−1. (5)(6)

Similar to studying the unit root in continuous time series, we are only concerned with statistical properties of the autoregressive coefficient estimator. For the nonstationary of continuous-valued time series, we often need to examine whether the characteristic polynomial of AR(1) process has a unit root. Thus, we want to see if we can find the limiting distribution of the autoregressive coefficient estimator. Let us present a result that is needed later on.

Lemma 3. Suppose that 𝑍_𝑡 follows a random walk without drift, 𝑍_𝑡=𝑍_𝑡−1+𝑤_𝑡, (7) where 𝑍₀=0 and {𝑤_𝑡} is an i.i.d. sequence with mean zero and variance 𝜎²_𝑤>0. Let “⇒” denote converges in distribution, and let 𝑊(𝑟) denote the standard Brownian motion. Then, one has the following properties:(i)𝑇^−1/2∑^𝑇_𝑡=1𝑤_𝑡⇒𝜎_𝑤𝑊(1), (ii)𝑇⁻¹∑^𝑇_𝑡=1𝑍_𝑡−1𝑤_𝑡⇒(1/2)𝜎²_𝑤[𝑊²(1)−1], (iii)𝑇^−3/2∑^𝑇_𝑡=1𝑡𝑤_𝑡⇒𝜎_𝑤𝑊(1)−𝜎_𝑤∫¹₀𝑊(𝑟)𝑑𝑟, (iv)𝑇^−3/2∑^𝑇_𝑡=1𝑍_𝑡−1⇒𝜎_𝑤∫¹₀𝑊(𝑟)𝑑𝑟, (v)𝑇^−5/2∑^𝑇_𝑡=1𝑡𝑍_𝑡−1⇒𝜎_𝑤∫¹₀𝑟𝑊(𝑟)𝑑𝑟, (vi)𝑇⁻²∑^𝑇_𝑡=1𝑍²_𝑡−1⇒𝜎²_𝑤∫¹₀𝑊²(𝑟)𝑑𝑟, (vii)𝑇^−(𝑛+1)∑^𝑇_𝑡=1𝑡^𝑛→1/(𝑛+1), for 𝑛=0,1,2,….

Proof. See Proposition 17.1 in Hamilton [26].

Theorem 4. If all the assumptions of Definition 1 hold, then one has(i)𝑇⁻¹∑^𝑇_𝑡=1𝜀_𝑡(𝜃_𝑘∘𝜀_𝑡−𝑘) converges in mean square to 𝜃_𝑘𝜇²_𝜀, for 𝑘=1,…,𝑞,(ii)𝑇⁻¹∑^𝑇_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)(𝜃_𝑗∘𝜀_𝑡−𝑗) converges in mean square to 𝜃_𝑖𝜃_𝑗𝜇²_𝜀, for 1≤𝑖<𝑗≤𝑞.

Proof. (i) First, we prove that 𝑇⁻¹∑^𝑇_𝑡=1𝜀_𝑡⋅(𝜃_𝑘∘𝜀_𝑡−𝑘) is integrable in mean square. Using the well-known results: 𝐸𝑋(𝑋)𝐸(𝑋)𝐸(𝑌),𝐸𝛽∘𝑌=𝐸𝛽∘𝑌=𝛽𝐸𝛽∘𝑌²=𝐸𝐸𝛽∘𝑌²𝐸(𝑌)∣𝑌=𝛽1−𝛽+𝛽²𝐸𝑌², (8) where 𝑋 and 𝑌 are independent and 𝛽∈[0,1], we can derive that 𝐸𝑇^𝑇−1_𝑡=1𝜀_𝑡𝜃_𝑘∘𝜀_𝑡−𝑘²=𝑇^−2𝑇_𝑡=1𝐸𝜀²_𝑡𝜃_𝑘∘𝜀_𝑡−𝑘²+2_{1≤𝑚<𝑛≤𝑇}𝐸𝜀_𝑚⋅𝜃_𝑘∘𝜀_𝑚−𝑘𝜀_𝑛⋅𝜃_𝑘∘𝜀_𝑛−𝑘=𝑇⁻²𝑇𝜇²_𝜀+𝜎²_𝜀𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜇²_𝜀+𝜎²_𝜀+𝑇²−𝑇𝜃²_𝑘𝜇⁴_𝜀=𝑇⁻¹𝜇²_𝜀+𝜎²_𝜀𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜇²_𝜀+𝜎²_𝜀+1−𝑇⁻¹𝜃²_𝑘𝜇⁴_𝜀≤𝜇²_𝜀+𝜎²_𝜀𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜇²_𝜀+𝜎²_𝜀+𝜃²_𝑘𝜇⁴_𝜀<∞. (9) Therefore, 𝑇⁻¹∑^𝑇_𝑡=1𝜀_𝑡⋅(𝜃_𝑘∘𝜀_𝑡−𝑘) is integrable.
Next, we show that it converges to 𝜃_𝑘𝜇²_𝜀 in mean square. In fact, 𝐸𝑇^𝑇−1_𝑡=1𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘−𝜃_𝑘𝜇²_𝜀²=𝐸𝑇^𝑇−1_𝑡=1𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘−𝑇^𝑇−1_𝑡=1𝐸(𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘)²=𝑇^𝑇−2_𝑡=1𝐸𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘²−𝐸𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘²+2𝑇^{−2𝑇−𝑘}_𝑡=𝑘+1cov𝜃_𝑘∘𝜀_𝑡−𝑘⋅𝜀_𝑡,𝜃_𝑘∘𝜀_𝑡⋅𝜀_𝑡+𝑘=𝑇⁻¹𝜇²_𝜀+𝜎²_𝜀𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜇²_𝜀+𝜎²_𝜀−𝜃²_𝑘𝜇⁴_𝜀+2𝑇⁻²(𝑇−2𝑘)𝜃²_𝑘𝜇²_𝜀𝜎²_𝜀. (10)
From the assumptions 𝜇_𝜀<∞,  𝜎²_𝜀<∞, and 𝜃_𝑘∈[0,1], we get lim_𝑇→∞𝐸(𝑇⁻¹∑^𝑇_𝑡=1𝜀_𝑡⋅(𝜃_𝑘∘𝜀_𝑡−𝑘)−𝜃_𝑘𝜇²_𝜀)²=0.
This completes the proof for (i).
(ii) Without loss of generality, we assume that 1≤𝑖<𝑗≤𝑞. We have 𝐸𝑇^𝑇−1_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)⋅(𝜃_𝑗∘𝜀_𝑡−𝑗)²=𝑇^𝑇−2_𝑡=𝑗𝐸𝜃_𝑖∘𝜀_𝑡−𝑖𝜃_𝑗∘𝜀_𝑡−𝑗²+2𝑇⁻²_{1≤𝑚<𝑛≤𝑇}𝐸𝜃_𝑖∘𝜀_𝑚−𝑖⋅𝜃_𝑗∘𝜀_𝑚−𝑗𝜃_𝑖∘𝜀_𝑛−𝑖⋅𝜃_𝑗∘𝜀_𝑛−𝑗=𝑇⁻¹𝜃_𝑖1−𝜃_𝑖𝜇_𝜀+𝜃²_𝑖𝜇²_𝜀+𝜎²_𝜀×𝜃_𝑗1−𝜃_𝑗𝜇_𝜀+𝜃²_𝑗𝜇²_𝜀+𝜎²_𝜀+𝑇⁻²12𝑇−𝑗−1𝑇−𝑗𝜃²_𝑖𝜃²_𝑗𝜇⁴_𝜀≤𝜃_𝑖1−𝜃_𝑖𝜇_𝜀+𝜃²_𝑖𝜇²_𝜀+𝜎²_𝜀×𝜃_𝑗1−𝜃_𝑗𝜇_𝜀+𝜃²_𝑗𝜇²_𝜀+𝜎²_𝜀+12𝜃²_𝑖𝜃²_𝑗𝜇⁴_𝜀<∞. (11)
Thus, 𝑇⁻¹∑^𝑇_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)⋅(𝜃_𝑗∘𝜀_𝑡−𝑗) is integrable.
Next, we prove that the limit holds lim_𝑇→∞𝐸(𝑇⁻¹∑^𝑇_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)⋅(𝜃_𝑗∘𝜀_𝑡−𝑗)−𝜃_𝑖𝜃_𝑗𝜇²_𝜀)²=0, 𝐸𝑇^𝑇−1_𝑡=1𝜃_𝑖∘𝜀_𝑡−𝑖⋅𝜃_𝑗∘𝜀_𝑡−𝑗−𝜃_𝑖𝜃_𝑗𝜇²_𝜀²=𝐸𝑇^𝑇−1_𝑡=1𝜃_𝑖∘𝜀_𝑡−𝑖⋅𝜃_𝑗∘𝜀_𝑡−𝑗−𝑇^𝑇−1_𝑡=1𝐸𝜃_𝑖∘𝜀_𝑡−𝑖⋅𝜃_𝑗∘𝜀_𝑡−𝑗²=𝑇^𝑇−2_𝑡=1Var𝜃_𝑖∘𝜀_𝑡−𝑖𝜃_𝑗∘𝜀_𝑡−𝑗+2𝑇⁻²_{1≤𝑚<𝑛≤𝑇}cov𝜃_𝑖∘𝜀_𝑚−𝑖⋅𝜃_𝑗∘𝜀_𝑚−𝑗,𝜃_𝑖∘𝜀_𝑛−𝑖⋅𝜃_𝑗∘𝜀_𝑛−𝑗=𝑇^𝑇−2_𝑡=1Var𝜃_𝑖∘𝜀_𝑡−𝑖𝜃_𝑗∘𝜀_𝑡−𝑗+𝑇^{−2𝑇−𝑖+𝑗}_{𝑘=max{1,1−𝑖+𝑗}}cov𝜃_𝑖∘𝜀_𝑘−𝑖⋅𝜃_𝑗∘𝜀_𝑘−𝑗,𝜃_𝑖∘𝜀_{𝑘−2𝑖+𝑗}⋅𝜃_𝑗∘𝜀_𝑘−𝑖=𝑇⁻¹𝜃_𝑖1−𝜃_𝑖𝜇_𝜀+𝜃²_𝑖𝜇²_𝜀+𝜎²_𝜀×𝜃_𝑗1−𝜃_𝑗𝜇_𝜀+𝜃²_𝑗𝜇²_𝜀+𝜎²_𝜀+𝑇⁻²𝑇−𝑖+𝑗−max1,1−𝑖+𝑗×𝜃²_𝑖𝜃_𝑗1−𝜃_𝑗𝜇³_𝜀+𝜃²_𝑖𝜃²_𝑗𝜇²_𝜀𝜎²_𝜀. (12)
By using 𝜇_𝜀<∞,  𝜎²_𝜀<∞, and 𝜃_𝑘∈[0,1], with the above arguments, we get lim_𝑇→∞𝐸𝑇^𝑇−1_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)(𝜃_𝑗∘𝜀_𝑡−𝑗)−𝜃_𝑖𝜃_𝑗𝜇²_𝜀²=0. (13) Then, 𝑇⁻¹∑^𝑇_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)(𝜃_𝑗∘𝜀_𝑡−𝑗) converges in mean square to 𝜃_𝑖𝜃_𝑗𝜇²_𝜀.

Theorem 5. If the process 𝑋_𝑡 is defined as in Definition 1, then one has(i)𝑇⁻¹∑^𝑇_𝑡=1𝑢_𝑡⇒𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘), (ii)𝑇⁻²∑^𝑇_𝑡=1𝑋_𝑡−1𝑢_𝑡⇒(1/2)𝜇²_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘)², (iii)𝑇⁻²∑^𝑇_𝑡=1𝑡𝑢_𝑡⇒(1/2)𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘), (iv)𝑇⁻²∑^𝑇_𝑡=1𝑋_𝑡−1⇒(1/2)𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘), (v)𝑇⁻³∑^𝑇_𝑡=1𝑡𝑋_𝑡−1⇒(1/3)𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘), (vi)𝑇⁻³∑^𝑇_𝑡=1𝑋²_𝑡−1⇒(1/3)𝜇²_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘)².

Proof. Let 𝜀^∗_𝑡=𝜀_𝑡−𝜇_𝜀, 𝜀^∗_𝑘,𝑡=𝜃_𝑘∘𝜀_𝑡−𝑘−𝜃_𝑘𝜇_𝜀, 𝑘=1,…,𝑞.
Then, we have the means and variances of 𝜀^∗_𝑡and 𝜀^∗_𝑘,𝑡 given by 𝐸𝜀^∗_𝑡=0,𝜎_𝜀^∗_𝑡=Var𝜀^∗_𝑡=𝜎_𝜀,𝐸𝜀^∗_𝑘,𝑡=0,𝜎_𝜀^∗_𝑘,𝑡=Var𝜀^∗_𝑘,𝑡=𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜎²_𝜀. (14) It is easy to see that {𝜀^∗_𝑡} is a sequence of i.i.d. random variables. For a fixed 𝑘, {𝜀^∗_𝑘,𝑡} is also a sequence of i.i.d. random variables.
(i) Straightforward using the law of large numbers.
(ii) One has ^𝑇_𝑡=1𝑋_𝑡−1𝑢_𝑡=𝑋₀𝑢₁+𝑋₁𝑢₂+⋯+𝑋_𝑇−1𝑢_𝑇=0+𝑢₁𝑢₂+𝑢₁+𝑢₂𝑢₃+⋯+𝑢₁+⋯+𝑢_𝑇−1𝑢_𝑇=𝑢₁𝑢₂+⋯+𝑢_𝑇+𝑢₂𝑢₃+⋯+𝑢_𝑇+⋯+𝑢_𝑇−1𝑢_𝑇=_𝑖<𝑗𝑢_𝑖𝑢_𝑗=12^𝑇_𝑡=1𝑢_𝑡²−^𝑇_𝑡=1𝑢²_𝑡. (15)
From the conclusion in (i), we get 𝑇^𝑇−1_𝑡=1𝑢_𝑡²⟹𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘². (16)
Note that ^𝑇_𝑡=1𝑢²_𝑡=^𝑇_𝑡=1𝜀²_𝑡+^𝑞_𝑘=1^𝑇_𝑡=1𝜃_𝑘∘𝜀_𝑡−𝑘²𝜇_𝑋+^𝑞_𝑘=1^𝑇_𝑡=1𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘+2_{1≤𝑖<𝑗≤𝑞}^𝑇_𝑡=1𝜃_𝑖∘𝜀_𝑡−𝑖⋅𝜃_𝑗∘𝜀_𝑡−𝑗. (17)
Using the law of large numbers, we obtain 𝑇^𝑇−1_𝑡=1𝜀²_𝑡⟹𝐸𝜀²_𝑡=𝜇²_𝜀+𝜎²_𝜀,𝑇^𝑇−1_𝑡=1𝜃_𝑘∘𝜀_𝑡−𝑘²⟹𝐸𝜃_𝑘∘𝜀_𝑡−𝑘²=𝜃_𝑘1−𝜃_𝑘𝜇_𝜀+𝜃²_𝑘𝜇²_𝜀+𝜎²_𝜀=𝜎²_𝜀^∗_𝑘,𝑡,𝑘=1,…,𝑞. (18)
Recall Theorem 4, where 𝑇⁻¹∑^𝑇_𝑡=1𝜀_𝑡(𝜃_𝑘∘𝜀_𝑡−𝑘) converges in mean square to 𝜃_𝑘𝜇²_𝜀 and 𝑇⁻¹∑^𝑇_𝑡=1(𝜃_𝑖∘𝜀_𝑡−𝑖)(𝜃_𝑗∘𝜀_𝑡−𝑗) converges in mean square to 𝜃_𝑖𝜃_𝑗𝜇²_𝜀, and thus 𝑇^𝑇−1_𝑡=1𝜀_𝑡⋅𝜃_𝑘∘𝜀_𝑡−𝑘⟹𝜃_𝑘𝜇²_𝜀,𝑇^𝑇−1_𝑡=1𝜃_𝑖∘𝜀_𝑡−𝑖𝜃_𝑗∘𝜀_𝑡−𝑗⟹𝜃_𝑖𝜃_𝑗𝜇²_𝜀. (19)
Then, we get 𝑇⁻¹∑^𝑇_𝑡=1𝑢²_𝑡⇒(𝜇²_𝜀+𝜎²_𝜀)+∑^𝑞_𝑘=1𝜎²_𝜀^∗_𝑘,𝑡+2(∑^𝑞_𝑘=1𝜃_𝑘+∑_{1≤𝑖<𝑗≤𝑞}𝜃_𝑖𝜃_𝑗)𝜇²_𝜀.
Therefore, 𝑇^𝑇−2_𝑡=1𝑋_𝑡−1𝑢_𝑡=12𝑇^𝑇−1_𝑡=1𝑢_𝑡²−𝑇⁻¹𝑇^𝑇−1_𝑡=1𝑢²_𝑡⟹12𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘²=1−02𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘². (20)
(iii) Morever, 𝑇^𝑇−2_𝑡=1𝑡𝑢_𝑡=𝑇^𝑇−2_𝑡=1𝑡𝜀_𝑡+𝜃₁∘𝜀_𝑡−1+⋯+𝜃_𝑞∘𝜀_𝑡−𝑞=𝑇^𝑇−2_𝑡=1𝑡𝜀^∗_𝑡+𝜇_𝜀+^𝑞_𝑘=1𝑇^𝑇−2_𝑡=1𝑡𝜀^∗_𝑘,𝑡+𝜃_𝑘𝜇_𝜀=𝑇^−1/2𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡+^𝑞_𝑘=1𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑘,𝑡+𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘𝑇^𝑇−2_𝑡=1𝑡. (21)
From (iii) and (vii) of Lemma 3, we have 𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡⟹𝜎_𝜀^∗_𝑡𝑊(1)−𝜎_𝜀^∗_𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,_{𝑇−3/2𝑡=1}𝑡𝜀^∗_𝑘,𝑡⟹𝜎_𝜀^∗_𝑘,𝑡𝑊(1)−𝜎_𝜀^∗_𝑘,𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,𝑘=1,…,𝑞,_{𝑇−2𝑡=1}1𝑡⟶2. (22)
Therefore, 𝑇⁻²∑^𝑇_𝑡=1𝑡𝑢_𝑡⇒0+0+(1/2)𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘)=(1/2)𝜇_𝜀(1+∑^𝑞_𝑘=1𝜃_𝑘). Consider the following:
(iv) 𝑇^𝑇−2_𝑡=1𝑋_𝑡−1=𝑇^−2𝑇−1_𝑡=1(𝑇−𝑡)𝑢_𝑡=𝑇^−2𝑇−1_𝑡=1(𝑇−𝑡)𝜀_𝑡+𝜃₁∘𝜀_𝑡−1+⋯+𝜃_𝑞∘𝜀_𝑡−𝑞=𝑇^𝑇−2_𝑡=1(𝑇−𝑡)𝜀^∗_𝑡+𝜇_𝜀+𝑇^𝑞−2𝑇_𝑘=1𝑡=1(𝑇−𝑡)𝜀^∗_𝑘,𝑡+𝜃_𝑘𝜇_𝜀=𝑇^𝑇−2_𝑡=1(𝑇−𝑡)𝜀^∗_𝑡+^𝑞_𝑘=1𝑇^𝑇−2_𝑡=1(𝑇−𝑡)𝜀^∗_𝑘,𝑡+𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘𝑇^𝑇−2_𝑡=1(𝑇−𝑡)=𝑇^𝑇−1_𝑡=1𝜀^∗_𝑡−𝑇^−1/2𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡+^𝑞_𝑘=1𝑇^𝑇−1_𝑡=1𝜀^∗_𝑘,𝑡−𝑇^−1/2𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑘,𝑡+𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘𝑇^𝑇−2_𝑡=1(𝑇−𝑡). (23)
By the law of large numbers, we have 𝑇^𝑇−1_𝑡=1𝜀^∗_𝑡𝑇⟹0,_{𝑇−1𝑡=1}𝜀^∗_𝑘,𝑡⟹0,𝑘=1,…,𝑞. (24)
From the (iii) and (vii) of Lemma 3, we get 𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡⟹𝜎_𝜀^∗_𝑡𝑊(1)−𝜎_𝜀^∗_𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,_{𝑇−3/2𝑡=1}𝑡𝜀^∗_𝑘,𝑡⟹𝜎_𝜀^∗_𝑘,𝑡𝑊(1)−𝜎_𝜀^∗_𝑘,𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,𝑘=1,…,𝑞,_{𝑇−2𝑡=1}()⟶1𝑇−𝑡2. (25)
Then, we have 𝑇^𝑇−2_𝑡=1𝑋_𝑡−11⟹0+0+0+2𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘=12𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘. (26)
(v) Elementary algebra gives us that 𝑋_𝑡=^𝑡_𝑘=1𝑢_𝑘=^𝑡_𝑘=1𝜀_𝑘+𝜃₁∘𝜀_𝑘−1+⋯+𝜃_𝑞∘𝜀_𝑘−𝑞=^𝑡_𝑘=1𝜀^∗_𝑘+^𝑡_𝑘=1𝜀^∗_1,𝑘+⋯+^𝑡_𝑘=1𝜀^∗_𝑞,𝑘+1+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀𝑡=𝜉_𝑡+^𝑞_𝑖=1𝜂_𝑖,𝑡+1+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀𝑡, (27) where 𝜉_𝑡=∑^𝑡_𝑘=1𝜀^∗_𝑘,𝜂_𝑖,𝑡=∑^𝑡_𝑘=1𝜀^∗_𝑖,𝑘,𝑖=1,…,𝑞 and, as assumed, 𝜉₀=𝜂_1,0=⋯=𝜂_𝑞,0=0. It is easy to see that 𝜉_𝑡 and 𝜂_𝑖,𝑡,  𝑖=1,…,𝑞, follow a random walk process without drift. Using (v) and (vii) of Lemma 3, we get 𝑇^𝑇−5/2_𝑡=1𝑡𝜉_𝑡−1⟹𝜎_𝜀^∗_𝑡¹₀(𝑟)𝑇𝑟𝑊𝑑𝑟,_{𝑇−5/2𝑡=1}𝑡𝜂_{𝑖,𝑡−1}⟹𝜎_𝜀^∗_𝑖,𝑡¹₀(𝑟)𝑇𝑟𝑊𝑑𝑟,𝑖=1,…,𝑞,_{𝑇−3𝑡=1}𝑡()⟶1𝑡−13. (28) Then, we get 𝑇^𝑇−3_𝑡=1𝑡𝑋_𝑡−1=𝑇^𝑇−3_𝑡=1𝑡𝜉_𝑡−1+^𝑞_𝑖=1𝑇^𝑇−3_𝑡=1𝑡𝜂_{𝑖,𝑡−1}+1+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀𝑇^𝑇−3_𝑡=1𝑡(𝑡−1)=𝑇^−1/2𝑇^𝑇−5/2_𝑡=1𝑡𝜉_𝑡−1+𝑇^𝑞−1/2_𝑖=1𝑇^𝑇−5/2_𝑡=1𝑡𝜂_{𝑖,𝑡−1}+1+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀𝑇^𝑇−3_𝑡=1𝑡(1𝑡−1)⟹0+0+31+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀=131+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀. (29)
(vi) One has 𝑇^𝑇−3_𝑡=1𝑋²_𝑡−1=𝑇^𝑇−3_𝑡=1𝜉_𝑡−1+^𝑞_𝑖=1𝜂_{𝑖,𝑡−1}+1+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀(𝑡−1)²=𝑇^𝑇−3_𝑡=1𝜉²_𝑡−1+𝑇^𝑞−3𝑇_𝑖=1𝑡=1𝜂²_{𝑖,𝑡−1}+1+^𝑞_𝑘=1𝜃_𝑘²𝜇²_𝜀𝑇^𝑇−3_𝑡=1(𝑡−1)²+2^𝑞_𝑖=1𝑇^𝑇−3_𝑡=1𝜉_𝑡−1𝜂_{𝑖,𝑡−1}+21+^𝑞_𝑘=1𝜃_𝑘𝜇_𝜀𝑇^𝑇−3_𝑡=1𝜉_𝑡−1+^𝑞_𝑖=1𝜂_{𝑖,𝑡−1}(𝑡−1). (30)
Firstly, we prove 𝑇⁻³∑^𝑇_𝑡=1(𝜉_𝑡−1+∑^𝑞_𝑖=1𝜂_{𝑖,𝑡−1})(𝑡−1)⇒0.
By (iii) and (v) of Lemma 3, we have 𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡⟹𝜎_𝜀^∗_𝑡𝑊(1)−𝜎_𝜀^∗_𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,_{𝑇−5/2𝑡=1}𝑡𝜉_𝑡−1⟹𝜎_𝜀^∗_𝑡¹₀𝑟𝑊(𝑟)𝑑𝑟. (31)
It is easy to see that 𝑇⁻²𝜉_𝑇−1⇒0 and 𝑇⁻²𝜀^∗_𝑇⇒0.
Thus, 𝑇^𝑇−3_𝑡=1(𝑡−1)𝜉_𝑡−1=𝑇^−3𝑇−1_𝑡=1𝑡𝜉_𝑡=𝑇^−3𝑇−1_𝑡=1𝑡𝜉_𝑡−1+𝜀^∗_𝑡=𝑇^𝑇−3_𝑡=1𝑡𝜉_𝑡−1+𝜀^∗_𝑡−𝑇⁻²𝜉_𝑇−1−𝑇⁻²𝜀^∗_𝑇=𝑇^−1/2𝑇^{−5/2𝑇−1}_𝑡=1𝑡𝜉_𝑡−1+𝑇^−3/2𝑇^{−3/2𝑇−1}_𝑡=1𝑡𝜀^∗_𝑡−𝑇⁻²𝜉_𝑇−1−𝑇⁻²𝜀^∗_𝑇⟹0. (32)
By using a similar approach, we get 𝑇⁻³∑^𝑇_𝑡=1(𝑡−1)𝜂_{𝑖,𝑡−1}⇒0,  𝑖=1,…,𝑞.
Therefore, 𝑇^𝑇−3_𝑡=1𝜉_𝑡−1+^𝑞_𝑖=1𝜂_{𝑖,𝑡−1}(𝑡−1)=𝑇^𝑇−3_𝑡=1(𝑡−1)𝜉_𝑡−1+^𝑞_𝑖=1𝑇^𝑇−3_𝑡=1(𝑡−1)𝜂_{𝑖,𝑡−1}⟹0+0=0. (33)
Secondly, we prove that the limit 𝑇⁻³∑^𝑇_𝑡=1𝜉_𝑡−1𝜂_{𝑖,𝑡−1}⇒0,  𝑖=1,…,𝑞, holds.
Using the well-known inequality |𝜉_𝑡−1𝜂_{𝑖,𝑡−1}|≤(1/2)(𝜉²_𝑡−1+𝜂²_{𝑖,𝑡−1}), we find that 𝑇^𝑇−3_𝑡=1𝜉_𝑡−1𝜂_{𝑖,𝑡−1}≤𝑇^𝑇−3_𝑡=1𝜉_𝑡−1𝜂_{𝑖,𝑡−1}≤12𝑇^𝑇−3_𝑡=1𝜉²_𝑡−1+𝑇^𝑇−3_𝑡=1𝜂²_{𝑖,𝑡−1}. (34)
From (vi) of Lemma 3, we get 𝑇^𝑇−2_𝑡=1𝜉²_𝑡−1⟹𝜎²_𝜀^∗_𝑡¹₀𝑊²(𝑟)𝑇𝑑𝑟,_{𝑇−2𝑡=1}𝜂²_{𝑖,𝑡−1}⟹𝜎²_𝜀^∗_𝑖,𝑡¹₀𝑊²(𝑟)𝑑𝑟. (35)
The two limits imply that 𝑇^𝑇−3_𝑡=1𝜉²_𝑡−1⟹0,𝑇^𝑇−3_𝑡=1𝜂²_{𝑖,𝑡−1}⟹0. (36)
By the Cauchy-Schwarz theorem, we obtain 𝑇⁻³∑^𝑇_𝑡=1𝜉_𝑡−1𝜂_{𝑖,𝑡−1}⇒0,  𝑖=1,…,𝑞.
From (iii), (vi), and (vii) of Lemma 3, we obtain 𝑇^𝑇−2_𝑡=1𝜉²_𝑡−1⟹𝜎²_𝜀^∗_𝑡¹₀𝑊²(𝑟)𝑇𝑑𝑟,_{𝑇−2𝑡=1}𝜂²_{𝑖,𝑡−1}⟹𝜎²_𝜀^∗_𝑖,𝑡¹₀𝑊²(𝑟)𝑇𝑑𝑟,_{𝑇−3𝑡=1}(𝑡−1)²⟶13,𝑇^𝑇−3/2_𝑡=1𝑡𝜀^∗_𝑡⟹𝜎_𝜀^∗_𝑡𝑊(1)−𝜎_𝜀^∗_𝑡¹₀𝑊(𝑟)𝑇𝑑𝑟,_{𝑇−3/2𝑡=1}𝑡𝜀^∗_𝑖,𝑡⟹𝜎_𝜀^∗_𝑖,𝑡𝑊(1)−𝜎_𝜀^∗_𝑖,𝑡¹₀𝑊(𝑟)𝑑𝑟. (37)
Therefore, 𝑇⁻³∑^𝑇_𝑡=1𝑋²_𝑡−1⇒0+0+(1/3)(1+∑^𝑞_𝑘=1𝜃_𝑘)²𝜇²_𝜀+0+0=(1/3)(1+∑^𝑞_𝑘=1𝜃_𝑘)²𝜇²_𝜀.
The proof of this theorem is complete.

Theorem 6. The conditional least squares estimators of 𝛼 given by (5) converges in distribution to constant 1, when the true process is a nonstationary INAR(1) model with INMA(𝑞) innovation.

Proof. We first derive the numerator limit of (5): 𝑇⁻⁴𝑇^𝑇_𝑡=1𝑋_𝑡−1𝑋_𝑡−^𝑇_𝑡=1𝑋_𝑡−1^𝑇_𝑡=1𝑋_𝑡=𝑇^𝑇−3_𝑡=1𝑋_𝑡−1𝑋_𝑡−𝑇^𝑇−2_𝑡=1𝑋_𝑡−1𝑇^𝑇−2_𝑡=1𝑋_𝑡=𝑇^𝑇−3_𝑡=1𝑋²_𝑡−1+𝑇⁻¹𝑇^𝑇−2_𝑡=1𝑋_𝑡−1𝑢_𝑡−𝑇^𝑇−2_𝑡=1𝑋_𝑡−1𝑇^𝑇−2_𝑡=1𝑋_𝑡. (38)
By the (ii), (iv), and (vi) of Theorem 5, we have 𝑇^𝑇−2_𝑡=1𝑋_𝑡−1𝑢_𝑡⟹12𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘²,𝑇^𝑇−2_𝑡=1𝑋_𝑡−1⟹12𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘,𝑇^𝑇−2_𝑡=1𝑋_𝑡=𝑇^𝑇−2_𝑡=1𝑋_𝑡−1+𝑇⁻¹𝑇^𝑇−1_𝑡=1𝑢_𝑡⟹12𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘,𝑇^𝑇−3_𝑡=1𝑋²_𝑡−1⟹13𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘². (39) Then, we get 𝑇⁻⁴𝑇^𝑇_𝑡=1𝑋_𝑡−1𝑋_𝑡−^𝑇_𝑡=1𝑋_𝑡−1^𝑇_𝑡=1𝑋_𝑡⟹13𝜇²_𝜀1+^𝑞_𝑘=1𝜃_𝑘²1+0−2𝜇_𝜀1+^𝑞_𝑘=1𝜃_𝑘²=1𝜇12^2𝜀1+^𝑞_𝑘=1𝜃_𝑘². (40) Similarly, we have the denominator limit of (5): 𝑇⁻⁴𝑇^𝑇_𝑡=1𝑋²_𝑡−1−^𝑇_𝑡=1𝑋_𝑡−1²⟹1𝜇12^2𝜀1+^𝑞_𝑘=1𝜃_𝑘². (41) Using the Slutsky theorem, we obtain 𝛼=𝑇∑^𝑇_𝑡=1𝑋_𝑡−1𝑋_𝑡−∑^𝑇_𝑡=1𝑋_𝑡−1∑^𝑇_𝑡=1𝑋_𝑡𝑇∑^𝑇_𝑡=1𝑋²_𝑡−1−∑^𝑇_𝑡=1𝑋_𝑡−1²=𝑇⁻⁴𝑇∑^𝑇_𝑡=1𝑋_𝑡−1𝑋_𝑡−∑^𝑇_𝑡=1𝑋_𝑡−1∑^𝑇_𝑡=1𝑋_𝑡𝑇⁻⁴𝑇∑^𝑇_𝑡=1𝑋²_𝑡−1−∑^𝑇_𝑡=1𝑋_𝑡−1²⟹(1/12)𝜇²_𝜀1+∑^𝑞_𝑘=1𝜃_𝑘²(1/12)𝜇²_𝜀1+∑^𝑞_𝑘=1𝜃_𝑘²=1. (42) This completes the proof.

4. Simulation Study

To study the empirical performance of the CLS estimator of the autoregressive coefficient for an auxiliary regression process, while the true process is NSINARMA(1,𝑞) process, a brief simulation study is conducted.

Consider the true process, 𝑋_𝑡=𝑋_𝑡−1+𝑢_𝑡,𝑢_𝑡=𝜀_𝑡+𝜃₁∘𝜀_𝑡−1+⋯+𝜃_𝑞∘𝜀_𝑡−𝑞, (43) where {𝜀_𝑡} is an an i.i.d. sequence of the Poisson random variables with parameter 𝜆=0.3,5. The lag orders and coefficient parameters values considered are(i)for 𝑞=1,  𝜃₁∈{0.1,0.4,0.7,0.9},(ii)for 𝑞=2, (𝜃₁,𝜃₂)∈{(0.1,0.1),(0.2,0.3),(0.1,0.6),(0.3,0.4),(0.4,0.4)},(iii)for 𝑞=3, (𝜃₁,𝜃₂,𝜃₃)∈{(0.1,0.1,0.1),(0.1,0.2,0.4),(0.2,0.1,0.2),(0.3,0.3,0.3),(0.5,0.3,0.1)}.The auxiliary regression process is an INAR(1) process, 𝑋_𝑡=𝛼∘𝑋_𝑡−1+𝑣_𝑡, (44) where {𝑣_𝑡} is a sequence of i.i.d. nonnegative integer-valued random variables. This simulation study is conducted to indicate the large sample performances of the CLS estimator of 𝛼 when the true process is NSINARMA(1,𝑞) process. In the simulation, we use 𝑋₀=𝜀₀=𝜀₋₁=⋯=𝜀_−𝑞=0. The study is based on 300 replications. For each replication, we estimate the model parameter 𝛼 and calculate the bias and MSE of the parameter estimates. The sample size is varied to be 𝑇 = 100, 300, and 800.

From the results reported in Tables 1, 2, and 3, we can see that CLS is a good estimation method. The estimates’ bias and MSE values are all small. Most of the biases are negative. All the bias and MSE values decrease with an increasing sample size 𝑇. When the coefficient 𝜃 of innovation process and the sample size are fixed, we find that the absolute values of bias and MSE become smaller with an increasing 𝜆. When the sample size is increased, the MSE and bias values both converge to zero. For example, the smallest bias and MSE values in the simulation showed in Table 1 are 9.7185𝑒−07 and 2.8335𝑒−10. This illustrates that the CLS estimator 𝛼 given by (5) converges to a constant.

Table 1 

Bias and MSE results of 𝛼 for NSINARMA(1,1) model.

Table 2 

Bias and MSE results of 𝛼 for NSINARMA(1,2) model.

Table 3 

Bias and MSE results of 𝛼 for NSINARMA(1,3) model.

5. Conclusions

In this paper, we have proposed a nonstationary INAR model for nonstationary integer-valued data. We have used an extended structure of the innovation process to allow the innovation with correlation to follow an INMA(𝑞) process. We have presented the moments and conditional moments for this model, proposed a CLS estimator for the autoregressive coefficient in auxiliary regression model, and obtained the asymptotic distribution for the estimator of coefficient. The simulation results indicate that our CLS method produces good estimates for large samples.

Acknowledgments

The authors are grateful to two anonymous referees for helpful comments and suggestions which greatly improved the paper. This work is supported by the National Natural Science Foundation of China (no. 71171166 and no. 71201126), the Science Foundation of Ministry of Education of China (no. 12XJC910001), and the Fundamental Research Funds for the Central Universities (no. JBK120405).