# Nonstationary INAR(1) Process withqth-Order Autocorrelation Innovation

Authors: Kaizhi Yu; Hong Zou; Daimin Shi
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is concerned with an integer-valued random walk process withqth-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.


---

## Body

## 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 Section2, 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{Xt} is said to be the NSINARMA(1,q) process if it satisfies the following recursive equations:
(1)Xt=Xt-1+ut,ut=εt+θ1∘εt-1+θ2∘εt-2+⋯+θq∘εt-q,
where {εt} is a sequence of i.i.d. nonnegative integer-valued random variables with finite mean με<∞, variance σε2<∞, and probability mass function fε. All counting series θk∘εt-k are mutually independent, and θk∈[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 ofXt hold.Proposition 2.
Fort≥1, one has(i)
E(XtXt-1)=Xt-1+(1+θ1+⋯+θq)με,(ii)
E(Xt)=t(1+θ1+⋯+θq)με,(iii)
Var(XtXt-1)=με∑k=1qθk(1-θk)+σε2(1+∑k=1qθk2),(iv)
Var(Xt)=t(με∑k=1qθk(1-θk)+σε2(1+∑k=1qθk2)).Proof.
It is straightforward to get (i) to (iii). We prove (iv) by induction with(2)Var(Xt)=Var(E(Xt∣Xt-1))+E(Var(Xt∣Xt-1))=Var(Xt-1)+(με∑k=1qθk(1-θk)+σε2(1+∑k=1qθk2))
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 processXt is an INAR(1) model,
(3)Xt=α∘Xt-1+vt,
where {vt} is a sequence of i.i.d. nonnegative integer-valued random variables with finite mean μv and variance σv2. 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(4)Q(β)=∑t=1T(Xt-α∘Xt-1-μv)2,
with β=(α,μv), be the CLS criterion function. The CLS estimators of α and μv are obtained by minimizing Q and are given by
(5)α^=T∑t=1TXt-1Xt-(∑t=1TXt-1)(∑t=1TXt)T∑t=1TXt-12-(∑t=1TXt-1)2,(6)μ^v=T-1(∑t=1TXt-α^∑t=1TXt-1).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 thatZt follows a random walk without drift,
(7)Zt=Zt-1+wt,
where Z0=0 and {wt} is an i.i.d. sequence with mean zero and variance σw2>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=1Twt⇒σwW(1),(ii)
T-1∑t=1TZt-1wt⇒(1/2)σw2[W2(1)-1],(iii)
T-3/2∑t=1Ttwt⇒σwW(1)-σw∫01W(r)dr,(iv)
T-3/2∑t=1TZt-1⇒σw∫01W(r)dr,(v)
T-5/2∑t=1TtZt-1⇒σw∫01rW(r)dr,(vi)
T-2∑t=1TZt-12⇒σw2∫01W2(r)dr,(vii)
T-(n+1)∑t=1Ttn→1/(n+1), for n=0,1,2,….Proof.
See Proposition17.1 in Hamilton [26].Theorem 4.
If all the assumptions of Definition1 hold, then one has(i)
T-1∑t=1Tεt(θk∘εt-k) converges in mean square to θkμε2, for k=1,…,q,(ii)
T-1∑t=1T(θi∘εt-i)(θj∘εt-j) converges in mean square to θiθjμε2, for 1≤i<j≤q.Proof.
(i) First, we prove thatT-1∑t=1Tεt·(θk∘εt-k) is integrable in mean square. Using the well-known results:
(8)E(X(β∘Y))=E(X)E(β∘Y)=βE(X)E(Y),E(β∘Y)2=E(E((β∘Y)2∣Y))=β(1-β)E(Y)+β2E(Y2),
where X and Y are independent and β∈[0,1], we can derive that
(9)E(T-1∑t=1Tεt(θk∘εt-k))2=T-2(∑1≤m<n≤TE(εm·θk∘εm-k)(εn·θk∘εn-k)∑t=1TE[εt2(θk∘εt-k)2]+2∑1≤m<n≤TE(εm·θk∘εm-k)(εn·θk∘εn-k))=T-2(T((με2+σε2)(θk2(με2+σε2)θk(1-θk)με=T-2(T((με2+σε2)+θk2(με2+σε2)))+(T2-T)θk2με4)=T-1((με2+σε2)(θk(1-θk)με+θk2(με2+σε2)))+(1-T-1)θk2με4≤((με2+σε2)(θk(1-θk)με+θk2(με2+σε2)))+θk2με4<∞.
Therefore, T-1∑t=1Tεt·(θk∘εt-k) is integrable.
Next, we show that it converges toθkμε2 in mean square. In fact,
(10)E(T-1∑t=1Tεt·θk∘εt-k-θkμε2)2=E(T-1∑t=1Tεt·θk∘εt-k-T-1∑t=1TE(εt·θk∘εt-k))2=T-2∑t=1T(E(εt·θk∘εt-k)2-(E(εt·θk∘εt-k))2)+2T-2∑t=k+1T-kcov(θk∘εt-k·εt,θk∘εt·εt+k)=T-1((με2+σε2)(θk2(με2+σε2)θk(1-θk)με=T-1((με2+mσε2)+θk2(με2+σε2))-θk2με4)+2T-2(T-2k)θk2με2σε2.
From the assumptionsμε<∞,  σε2<∞, and θk∈[0,1], we get limT→∞E(T-1∑t=1Tεt·(θk∘εt-k)-θkμε2)2=0.
This completes the proof for (i).
(ii) Without loss of generality, we assume that1≤i<j≤q. We have
(11)E(T-1∑t=1T(θi∘εt-i)·(θj∘εt-j))2=T-2∑t=jTE((θi∘εt-i)(θj∘εt-j))2+2T-2∑1≤m<n≤TE((θj∘εn-j)(θi∘εm-i)+2T-2∑1≤m<n≤TmE·(θj∘εm-j)(θi∘εn-i)+2T-2∑1≤m<n≤TmE·(θj∘εn-j))=T-1(θi(1-θi)με+θi2(με2+σε2))×(θj(1-θj)με+θj2(με2+σε2))+T-212(T-j-1)(T-j)θi2θj2με4≤(θi(1-θi)με+θi2(με2+σε2))×(θj(1-θj)με+θj2(με2+σε2))+12θi2θj2με4<∞.
Thus,T-1∑t=1T(θi∘εt-i)·(θj∘εt-j) is integrable.
Next, we prove that the limit holdslimT→∞E(T-1∑t=1T(θi∘εt-i)·(θj∘εt-j)-θiθjμε2)2=0,
(12)E(T-1∑t=1T(θi∘εt-i)·(θj∘εt-j)-θiθjμε2)2=E(T-1∑t=1T(θi∘εt-i)·(θj∘εt-j)-T-1∑t=1TE((θi∘εt-i)·(θj∘εt-j)))2=T-2∑t=1TVar((θi∘εt-i)(θj∘εt-j))+2T-2∑1≤m<n≤Tcov((θi∘εm-i)·(θj∘εm-j),+T-2∑k=max{1,1-i+j}T-i+jmm(θi∘εn-i)·(θj∘εn-j))=T-2∑t=1TVar((θi∘εt-i)(θj∘εt-j))+T-2∑k=max{1,1-i+j}T-i+jcov((θi∘εk-i)·(θj∘εk-j),+T-2∑k=max{1,1-i+j}T-i+jmmmm(θi∘εk-2i+j)·(θj∘εk-i))=T-1(θi(1-θi)με+θi2(με2+σε2))×(θj(1-θj)με+θj2(με2+σε2))+T-2(θi2θj(1-θj)με3(T-i+j-max{1,1-i+j})+nnT-2×θi2θj(1-θj)με3+θi2θj2με2σε2).
By usingμε<∞,  σε2<∞, and θk∈[0,1], with the above arguments, we get
(13)limT→∞E(T-1∑t=1T(θi∘εt-i)(θj∘εt-j)-θiθjμε2)2=0.
Then, T-1∑t=1T(θi∘εt-i)(θj∘εt-j) converges in mean square to θiθjμε2.Theorem 5.
If the processXt is defined as in Definition 1, then one has(i)
T-1∑t=1Tut⇒με(1+∑k=1qθk),(ii)
T-2∑t=1TXt-1ut⇒(1/2)με2(1+∑k=1qθk)2,(iii)
T-2∑t=1Ttut⇒(1/2)με(1+∑k=1qθk),(iv)
T-2∑t=1TXt-1⇒(1/2)με(1+∑k=1qθk),(v)
T-3∑t=1TtXt-1⇒(1/3)με(1+∑k=1qθk),(vi)
T-3∑t=1TXt-12⇒(1/3)με2(1+∑k=1qθk)2.Proof.
Letεt*=εt-με, εk,t*=θk∘εt-k-θkμε, k=1,…,q.
Then, we have the means and variances ofεt*and εk,t* given by
(14)E(εt*)=0,σεt*=Var(εt*)=σε,E(εk,t*)=0,σεk,t*=Var(εk,t*)=θk(1-θk)με+θk2σε2.
It is easy to see that {εt*} is a sequence of i.i.d. random variables. For a fixed k, {εk,t*} is also a sequence of i.i.d. random variables.
(i) Straightforward using the law of large numbers.
(ii) One has(15)∑t=1TXt-1ut=X0u1+X1u2+⋯+XT-1uT=0+u1u2+(u1+u2)u3+⋯+(u1+⋯+uT-1)uT=u1(u2+⋯+uT)+u2(u3+⋯+uT)+⋯+uT-1uT=∑i<juiuj=12((∑t=1Tut)2-∑t=1Tut2).
From the conclusion in (i), we get(16)(T-1∑t=1Tut)2⟹με2(1+∑k=1qθk)2.
Note that(17)∑t=1Tut2=∑t=1Tεt2+∑k=1q(∑t=1T(θk∘εt-k)2)μX+∑k=1q(∑t=1Tεt·(θk∘εt-k))+2∑1≤i<j≤q(∑t=1T(θi∘εt-i)·(θj∘εt-j)).
Using the law of large numbers, we obtain(18)T-1∑t=1Tεt2⟹E(εt2)=(με2+σε2),T-1∑t=1T(θk∘εt-k)2⟹E(θk∘εt-k)2=θk(1-θk)με+θk2(με2+σε2)=σεk,t*2,k=1,…,q.
Recall Theorem4, where T-1∑t=1Tεt(θk∘εt-k) converges in mean square to θkμε2 and T-1∑t=1T(θi∘εt-i)(θj∘εt-j) converges in mean square to θiθjμε2, and thus
(19)T-1∑t=1Tεt·(θk∘εt-k)⟹θkμε2,T-1∑t=1T(θi∘εt-i)(θj∘εt-j)⟹θiθjμε2.
Then, we getT-1∑t=1Tut2⇒(με2+σε2)+∑k=1qσεk,t*2+2(∑k=1qθk+∑1≤i<j≤qθiθj)με2.
Therefore,(20)T-2∑t=1TXt-1ut=12((T-1∑t=1Tut)2-T-1(T-1∑t=1Tut2))⟹12(με2(1+∑k=1qθk)2-0)=12με2(1+∑k=1qθk)2.
(iii) Morever,(21)T-2∑t=1Ttut=T-2∑t=1Tt(εt+θ1∘εt-1+⋯+θq∘εt-q)=T-2∑t=1Tt(εt*+με)+∑k=1q(T-2∑t=1Tt(εk,t*+θkμε))=T-1/2(T-3/2∑t=1Ttεt*+∑k=1q(T-3/2∑t=1Ttεk,t*))+με(1+∑k=1qθk)(T-2∑t=1Tt).
From (iii) and (vii) of Lemma3, we have
(22)T-3/2∑t=1Ttεt*⟹σεt*W(1)-σεt*∫01W(r)dr,T-3/2∑t=1Ttεk,t*⟹σεk,t*W(1)-σεk,t*∫01W(r)dr,T-3/2∑t=1Ttεk,t*⟹σεk,t*W(1)-bσεk,t*k=1,…,q,T-2∑t=1Tt⟶12.
Therefore,T-2∑t=1Ttut⇒0+0+(1/2)με(1+∑k=1qθk)=(1/2)με(1+∑k=1qθk). Consider the following:
(iv)(23)T-2∑t=1TXt-1=T-2∑t=1T-1(T-t)ut=T-2∑t=1T-1(T-t)(εt+θ1∘εt-1+⋯+θq∘εt-q)=T-2∑t=1T(T-t)(εt*+με)+T-2∑k=1q∑t=1T(T-t)(εk,t*+θkμε)=T-2∑t=1T(T-t)εt*+∑k=1q(T-2∑t=1T(T-t)εk,t*)+με(1+∑k=1qθk)(T-2∑t=1T(T-t))=T-1∑t=1Tεt*-T-1/2(T-3/2∑t=1Ttεt*)+∑k=1q(T-1∑t=1Tεk,t*-T-1/2(T-3/2∑t=1Ttεk,t*))+με(1+∑k=1qθk)(T-2∑t=1T(T-t)).
By the law of large numbers, we have(24)T-1∑t=1Tεt*⟹0,T-1∑t=1Tεk,t*⟹0,k=1,…,q.
From the (iii) and (vii) of Lemma3, we get
(25)T-3/2∑t=1Ttεt*⟹σεt*W(1)-σεt*∫01W(r)dr,T-3/2∑t=1Ttεk,t*⟹σεk,t*W(1)-σεk,t*∫01W(r)dr,T-3/2∑t=1Ttεk,t*⟹σεk,t*W(1)-σεk,t*k=1,…,q,T-2∑t=1T(T-t)⟶12.
Then, we have(26)T-2∑t=1TXt-1⟹0+0+0+12με(1+∑k=1qθk)=12με(1+∑k=1qθk).
(v) Elementary algebra gives us that(27)Xt=∑k=1tuk=∑k=1t(εk+θ1∘εk-1+⋯+θq∘εk-q)=∑k=1tεk*+∑k=1tε1,k*+⋯+∑k=1tεq,k*+(1+∑k=1qθk)μεt=ξt+∑i=1qηi,t+(1+∑k=1qθk)μεt,
where ξt=∑k=1tεk*,ηi,t=∑k=1tεi,k*,i=1,…,q and, as assumed, ξ0=η1,0=⋯=ηq,0=0. It is easy to see that ξt and ηi,t,  i=1,…,q, follow a random walk process without drift. Using (v) and (vii) of Lemma 3, we get
(28)T-5/2∑t=1Ttξt-1⟹σεt*∫01rW(r)dr,T-5/2∑t=1Ttηi,t-1⟹σεi,t*∫01rW(r)dr,i=1,…,q,T-3∑t=1Tt(t-1)⟶13.
Then, we get
(29)T-3∑t=1TtXt-1=T-3∑t=1Ttξt-1+∑i=1q(T-3∑t=1Ttηi,t-1)+(1+∑k=1qθk)με(T-3∑t=1Tt(t-1))=T-1/2(T-5/2∑t=1Ttξt-1)+T-1/2∑i=1q(T-5/2∑t=1Ttηi,t-1)+(1+∑k=1qθk)με(T-3∑t=1Tt(t-1))⟹0+0+13(1+∑k=1qθk)με=13(1+∑k=1qθk)με.
(vi) One has(30)T-3∑t=1TXt-12=T-3∑t=1T(ξt-1+∑i=1qηi,t-1+(1+∑k=1qθk)με(t-1))2=T-3∑t=1Tξt-12+T-3∑i=1q∑t=1Tηi,t-12+(1+∑k=1qθk)2με2(T-3∑t=1T(t-1)2)+2∑i=1q(T-3∑t=1Tξt-1ηi,t-1)+2(1+∑k=1qθk)με(T-3∑t=1T(ξt-1+∑i=1qηi,t-1)(t-1)).
Firstly, we proveT-3∑t=1T(ξt-1+∑i=1qηi,t-1)(t-1)⇒0.
By (iii) and (v) of Lemma3, we have
(31)T-3/2∑t=1Ttεt*⟹σεt*W(1)-σεt*∫01W(r)dr,T-5/2∑t=1Ttξt-1⟹σεt*∫01rW(r)dr.
It is easy to see thatT-2ξT-1⇒0 and T-2εT*⇒0.
Thus,(32)T-3∑t=1T(t-1)ξt-1=T-3∑t=1T-1tξt=T-3∑t=1T-1t(ξt-1+εt*)=T-3∑t=1Tt(ξt-1+εt*)-T-2ξT-1-T-2εT*=T-1/2(T-5/2∑t=1T-1tξt-1)+T-3/2(T-3/2∑t=1T-1tεt*)-T-2ξT-1-T-2εT*⟹0.
By using a similar approach, we getT-3∑t=1T(t-1)ηi,t-1⇒0,  i=1,…,q.
Therefore,(33)T-3∑t=1T(ξt-1+∑i=1qηi,t-1)(t-1)=T-3∑t=1T(t-1)ξt-1+∑i=1q(T-3∑t=1T(t-1)ηi,t-1)⟹0+0=0.
Secondly, we prove that the limitT-3∑t=1Tξt-1ηi,t-1⇒0,  i=1,…,q, holds.
Using the well-known inequality|ξt-1ηi,t-1|≤(1/2)(ξt-12+ηi,t-12), we find that
(34)|T-3∑t=1Tξt-1ηi,t-1|≤T-3∑t=1T|ξt-1ηi,t-1|≤12(T-3∑t=1Tξt-12+T-3∑t=1Tηi,t-12).
From (vi) of Lemma3, we get
(35)T-2∑t=1Tξt-12⟹σεt*2∫01W2(r)dr,T-2∑t=1Tηi,t-12⟹σεi,t*2∫01W2(r)dr.
The two limits imply that(36)T-3∑t=1Tξt-12⟹0,T-3∑t=1Tηi,t-12⟹0.
By the Cauchy-Schwarz theorem, we obtainT-3∑t=1Tξt-1ηi,t-1⇒0,  i=1,…,q.
From (iii), (vi), and (vii) of Lemma3, we obtain
(37)T-2∑t=1Tξt-12⟹σεt*2∫01W2(r)dr,T-2∑t=1Tηi,t-12⟹σεi,t*2∫01W2(r)dr,T-3∑t=1T(t-1)2⟶13,T-3/2∑t=1Ttεt*⟹σεt*W(1)-σεt*∫01W(r)dr,T-3/2∑t=1Ttεi,t*⟹σεi,t*W(1)-σεi,t*∫01W(r)dr.
Therefore,T-3∑t=1TXt-12⇒0+0+(1/3)(1+∑k=1qθk)2με2+0+0=(1/3)(1+∑k=1qθk)2με2.
The proof of this theorem is complete.Theorem 6.
The conditional least squares estimators ofα 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):
(38)T-4(T∑t=1TXt-1Xt-(∑t=1TXt-1)(∑t=1TXt))=T-3∑t=1TXt-1Xt-(T-2∑t=1TXt-1)(T-2∑t=1TXt)=T-3∑t=1TXt-12+T-1(T-2∑t=1TXt-1ut)-(T-2∑t=1TXt-1)(T-2∑t=1TXt).
By the (ii), (iv), and (vi) of Theorem5, we have
(39)T-2∑t=1TXt-1ut⟹12με2(1+∑k=1qθk)2,T-2∑t=1TXt-1⟹12με(1+∑k=1qθk),T-2∑t=1TXt=T-2∑t=1TXt-1+T-1(T-1∑t=1Tut)⟹12με(1+∑k=1qθk),T-3∑t=1TXt-12⟹13με2(1+∑k=1qθk)2.
Then, we get
(40)T-4(T∑t=1TXt-1Xt-(∑t=1TXt-1)(∑t=1TXt))⟹13με2(1+∑k=1qθk)2+0-(12με(1+∑k=1qθk))2=112με2(1+∑k=1qθk)2.
Similarly, we have the denominator limit of (5):
(41)T-4(T∑t=1TXt-12-(∑t=1TXt-1)2)⟹112με2(1+∑k=1qθk)2.
Using the Slutsky theorem, we obtain
(42)α^=T∑t=1TXt-1Xt-(∑t=1TXt-1)(∑t=1TXt)T∑t=1TXt-12-(∑t=1TXt-1)2=T-4(T∑t=1TXt-1Xt-(∑t=1TXt-1)(∑t=1TXt))T-4(T∑t=1TXt-12-(∑t=1TXt-1)2)⟹(1/12)με2(1+∑k=1qθk)2(1/12)με2(1+∑k=1qθk)2=1.
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,(43)Xt=Xt-1+ut,ut=εt+θ1∘εt-1+⋯+θq∘εt-q,
where {εt} 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)
forq=1,  θ1∈{0.1,0.4,0.7,0.9},(ii)
forq=2, (θ1,θ2)∈{(0.1,0.1),(0.2,0.3),(0.1,0.6),(0.3,0.4),(0.4,0.4)},(iii)
forq=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,
(44)Xt=α∘Xt-1+vt,
where {vt} 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 Tables1, 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 ofα for NSINARMA(1,1) model.
CLS
λ = 0.3
λ = 5
Sample size
100
300
800
100
300
800
θ
1
=
0.1
Bias(α)
-
2.3411
e
-
03
-
1.3253
e
-
04
-
2.4503
e
-
05
-
9.7709
e
-
05
1.2879
e
-
05
9.7185
e
-
07
MSE(α)
1.6443
e
-
03
5.2694
e
-
06
1.8012
e
-
07
2.8641
e
-
06
4.9761
e
-
08
2.8335
e
-
10
θ
1
=
0.4
Bias(α)
-
1.8837
e
-
03
-
6.2751
e
-
05
-
1.9073
e
-
05
-
9.3204
e
-
05
1.1088
e
-
05
2.1430
e
-
06
MSE(α)
1.0645
e
-
03
1.1813
e
-
06
1.0913
e
-
07
2.6061
e
-
06
3.6884
e
-
08
1.3778
e
-
09
θ
1
=
0.7
Bias(α)
-
1.3353
e
-
03
-
8.5133
e
-
05
-
1.3719
e
-
05
-
7.6709
e
-
05
2.1019
e
-
05
2.5092
e
-
06
MSE(α)
5.3488
e
-
04
2.1743
e
-
06
5.6460
e
-
08
1.7653
e
-
06
1.3254
e
-
07
1.8889
e
-
09
θ
1
=
0.9
Bias(α)
-
4.6077
e
-
04
-
6.1094
e
-
05
6.5116
e
-
06
-
2.1260
E
-
04
1.8841
E
-
05
1.3932
E
-
06
MSE(α)
6.3694
e
-
05
1.1197
e
-
06
1.2720
e
-
08
1.3559
E
-
05
1.0649
E
-
07
5.8233
E
-
10Table 2
Bias and MSE results ofα for NSINARMA(1,2) model.
CLS
λ = 0.3
λ = 5
Sample size
100
300
800
100
300
800
(
θ
1
,
θ
2
) = (0.1, 0.1)
Bias(α)
-
9.8012
e
-
04
-
8.0314
e
-
05
-
1.2149
e
-
05
1.6030
e
-
05
-
2.6319
e
-
05
-
7.9892
e
-
06
MSE(α)
2.8819
e
-
04
1.9351
e
-
06
4.4280
e
-
08
7.7085
e
-
07
2.0781
e
-
07
1.9148
e
-
08
(
θ
1
,
θ
2
) = (0.1, 0.6)
Bias(α)
-
9.0876
e
-
04
-
1.4515
e
-
04
3.6936
e
-
06
-
4.5047
e
-
05
-
3.6633
e
-
05
-
1.7588
e
-
06
MSE(α)
2.4775
e
-
04
6.3203
e
-
06
4.0928
e
-
09
6.0877
e
-
07
4.0259
e
-
07
9.2803
e
-
10
(
θ
1
,
θ
2
) = (0.2, 0.3)
Bias(α)
-
5.8582
e
-
04
-
8.2571
e
-
05
-
1.1720
e
-
05
4.7792
e
-
05
-
2.2042
e
-
05
-
7.8901
e
-
06
MSE(α)
1.0296
e
-
04
2.0454
e
-
06
4.1208
e
-
08
6.8522
e
-
07
1.4575
e
-
07
1.8676
e
-
08
(
θ
1
,
θ
2
) = (0.3, 0.4)
Bias(α)
-
1.0548
e
-
03
-
1.3550
e
-
04
-
2.1872
e
-
06
-
6.0986
e
-
05
-
2.1873
e
-
05
-
1.6658
e
-
08
MSE(α)
3.3379
e
-
04
5.5082
e
-
06
1.4352
e
-
09
1.1158
e
-
06
1.4353
e
-
07
8.3243
e
-
14
(
θ
1
,
θ
2
) = (0.4, 0.4)
Bias(α)
-
8.9921
e
-
04
-
1.4099
e
-
04
-
8.5553
e
-
07
-
6.6568
e
-
05
-
2.2816
e
-
05
1.0669
e
-
07
MSE(α)
2.4257
e
-
04
5.9633
e
-
06
2.1958
e
-
10
1.3294
e
-
06
1.5617
e
-
07
3.4148
e
-
12Table 3
Bias and MSE results ofα for NSINARMA(1,3) model.
CLS
λ = 0.3
λ = 5
Sample size
100
300
800
100
300
800
(
θ
1
,
θ
2
,
θ
3
) = (0.1, 0.1, 0.1)
Bias(α)
-
1.3742
e
-
03
-
1.4097
e
-
04
-
3.3819
e
-
05
-
4.4965
e
-
05
4.5510
e
-
06
-
5.1844
e
-
06
MSE(α)
5.6651
e
-
04
5.9616
e
-
06
3.4312
e
-
07
6.0655
e
-
07
6.2135
e
-
08
8.0635
e
-
09
(
θ
1
,
θ
2
,
θ
3
) = (0.1, 0.2, 0.4)
Bias(α)
-
2.0218
e
-
03
-
9.3641
e
-
05
8.2901
e
-
06
-
1.4846
e
-
04
-
8.0926
e
-
06
-
8.4087
e
-
07
MSE(α)
1.2263
e
-
03
2.6306
e
-
06
2.0618
e
-
08
6.6121
e
-
06
1.9647
e
-
08
2.1212
e
-
10
(
θ
1
,
θ
2
,
θ
3
) = (0.2, 0.1, 0.2)
Bias(α)
-
1.0825
e
-
03
-
1.1218
e
-
04
-
3.0775
e
-
05
-
2.0846
e
-
05
7.8758
e
-
06
-
4.3565
e
-
06
MSE(α)
3.5154
e
-
04
3.7751
e
-
06
2.8413
e
-
07
1.3037
e
-
07
1.8608
e
-
08
5.6936
e
-
09
(
θ
1
,
θ
2
,
θ
3
) = (0.3, 0.3, 0.3)
Bias(α)
-
1.6766
e
-
03
-
1.3506
e
-
04
8.8713
e
-
06
-
1.3902
e
-
04
-
2.6374
e
-
06
-
2.3441
e
-
07
MSE(α)
8.4330
e
-
04
5.4723
e
-
06
2.3610
e
-
08
5.7980
e
-
06
2.0868
e
-
09
1.6485
e
-
11
(
θ
1
,
θ
2
,
θ
3
) = (0.5, 0.3, 0.1)
Bias(α)
-
1.5676
e
-
03
-
1.3084
e
-
04
7.6103
e
-
06
-
1.5146
e
-
04
-
2.5173
e
-
06
-
2.9431
e
-
07
MSE(α)
7.3722
e
-
04
5.1355
e
-
06
1.7375
e
-
08
6.8819
e
-
06
1.9010
e
-
09
2.5985
e
-
11


## 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.



---

*Source: 951312-2013-04-16.xml*