It is frequent to encounter integer-valued time series which are small in value and show a trend having relatively large fluctuation. To handle such a matter, we present a new first order integer-valued moving average model process with structural changes. The models provide a flexible framework for modelling a wide range of dependence structures. Some statistical properties of the process are discussed and moment estimation is also given. Simulations are provided to give additional insight into the finite sample behaviour of the estimators.
1. Introduction
Integer-valued time series occur in many situations, often as counts of events in consecutive points of time, for example, the number of births at a hospital in successive months, the number of road accidents in a city in successive months, and big numbers even for frequently traded stocks. Integer-valued time series represent an important class of discrete-valued time series models. Because of the broad field of potential applications, a number of time series models for counts have been proposed in literature. McKenzie [1] introduced the first order integer-valued autoregressive, INAR(1), model. The statistical properties of the INAR(1) are discussed in McKenzie [2], Al-Osh and Alzaid [3]. The model is further generalized to a pth-order autoregression, INAR(p), by Alzaid and Al-Osh [4] and Du and Li [5]. The qth-order integer-valued moving average model, INMA(q), was introduced by Al-Osh and Alzaid [6] and in a slightly different form by McKenzie [7]. Ferland et al. [8] proposed an integer-valued GARCH model to study overdispersed counts, and Fokianos and Fried [9], Weiß [10], and Zhu and Wang [11–13] made further studies. Györfi et al. [14] proposed a nonstationary inhomogeneous INAR(1) process, where the autoregressive type coefficient slowly converges to one. Bakouch and Ristić [15] introduced a new stationary integer-valued autoregressive process of the first order with zero truncated Poisson marginal distribution. Kachour and Yao [16] introduced a class of autoregressive models for integer-valued time series using the rounding operator. Kim and Park [17] proposed an extension of integer-valued autoregressive INAR models by using a signed version of the thinning operator. Zheng et al. [18] proposed a first order random coefficient integer-valued autoregressive model and got its ergodicity, moments, and autocovariance functions of the process. Gomes and Canto e Castro [19] presented a random coefficient autoregressive process for count data based on a generalized thinning operator. Existence and weak stationarity conditions for these models were established. A simple bivariate integer-valued time series model with positively correlated geometric marginals based on the negative binomial thinning mechanism was presented by Ristić et al. [20], and some properties of the model are also considered. Pedeli and Karlis [21] considered a bivariate INAR(1) (BINAR(1)) process where cross correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations.
Structural changes in economic data frequently correspond to instabilities in the real world. However, most work in this area has been concentrated on models without structural changes. It seems that the integer-valued autoregressive moving average (INARMA) model with break point has not attracted too much attention. For instance, a new method for modelling the dynamics of rain sampled by a tipping bucket rain gauge was proposed by Thyregod et al. [22]. The models take the autocorrelation and discrete nature of the data into account. First order, second order, and threshold models are presented together with methods to estimate the parameters of each model. Monteiro et al. [23] introduced a class of self-exciting threshold integer-valued autoregressive models driven by independent Poisson-distributed random variables. Basic probabilistic and statistical properties of this class of models were discussed. Moreover, parameter estimation was also addressed. Hudecová [24] suggested a procedure for testing a change in the autoregressive models for binary time series. The test statistic is a maximum of normalized sums of estimated residuals from the model, and thus it is sensitive to any change which leads to a change in the unconditional success probability. Structural change is a statement about parameters, which only have meaning in the context of a model. In our discussion, we will focus on structural change in the simple count data model, the first order integer-valued moving average model, whose coefficient varies with the value of innovation. One of the leading reasons is that piecewise linear functions can offer a relatively simple approximation to the complex nonlinear dynamics.
The rest of this paper is divided into four sections. In Section 2, we give the definition and basic properties of the new INMA(1) model with structural changes. Section 3 discusses the estimation of the unknown parameters. We test the accuracy of the estimation via simulations in Section 4. Section 5 includes some concluding remarks.
2. Definition and Basic PropertiesDefinition 1.
Let {Xt} be a process with state space N0; let 0<αi<1, i=1,…,m, and τi, i=1,…,m-1, be positive integers. The process {Xt} is said to be first order integer-valued moving average model with structural change (INMASC(1)) if Xt satisfies the following equation:
(1)Xt={α1∘εt-1+εt,forτ0≤εt-1≤τ1α2∘εt-1+εt,forτ1<εt-1≤τ2vvvvvvv⋮αm-1∘εt-1+εt,forτm-2<εt-1≤τm-1αm∘εt-1+εt,forτm-1<εt-1<τm,
where {εt} is a sequence of independent and identically distributed Poisson random variables with mean λ and τ0∶=0, τm∶=∞.
The aim of this section is to provide expressions for the moments and stationary of INMASC(1) model. For this purpose, we introduce the following notations:
(2)pi∶=P(τi-1<εt≤τi),ui∶=E(τi-1<εt-1≤τi),σi2∶=Var(τi-1<εt-1≤τi),qi∶=1-pi,It-1,i∶={1,ifτi-1<εt-1≤τi0,otherwise,i=1,…,m.
Theorem 2.
The numerical characteristics of {Xt} are as follows:
(3)(i)μX∶=E(Xt)=∑i=1mpiαiui+λ,(ii)σX2∶=Var(Xt)=∑i=1mpiαi[αi(ui2+σi2)+(1-αi)ui]-(∑i=1mpiαiui)2+λ,(iii)γX(k)∶=cov(Xt,Xt-k)={∑i=1mpiαi(ui2+σi2-λui),k=10,k≥2.
Proof.
(i) It is easy to get the mean and variance of Xt by using the law of iterated expectations:
(4)E(Xt)=E[It-1,1(α1∘εt-1)+⋯+It-1,m(αm∘εt-1)+εt]=E{E[It-1,1(α1∘εt-1)vviiiiv+⋯+It-1,m(αm∘εt-1)∣εt-1]}+E(εt)=α1E(It-1,1εt-1)+⋯+αmE(It-1,mεt-1)+E(εt)=∑i=1mpiαiui+λ.
(ii) Moreover,
(5)Var(Xt)=Var(It-1,1(α1∘εt-1)+⋯+It-1,m(αm∘εt-1)+εt)=∑i=1mVar(It-1,i(αi∘εt-1))+Var(εt)+2∑i<jcov(It-1,i(αi∘εt-1),It-1,j(αj∘εt-1))=∑i=1m{Var(E[It-1,i(αi∘εt-1)∣εt-1])vv+E(Var[It-1,i(αi∘εt-1)∣εt-1])}+λ+2∑i<j{E[It-1,i(αi∘εt-1)It-1,j(αj∘εt-1)]vvvvvv-2E[It-1,i(αi∘εt-1)]E[It-1,j(αj∘εt-1)]}=∑i=1m[αi2Var(It-1,iεt-1)+αi(1-αi)E(It-1,iεt-1)]+λ-2∑i<jE{E[It-1,i(αi∘εt-1)∣εt-1]}vvvvvv×E{E[It-1,j(αj∘εt-1)∣εt-1]}=∑i=1m{[αi2E(It-12εt-12)-E2(It-1,iεt-1)]vv+piαi(1-αi)uiαi2E(It-12εt-12)-E2(It-1,iεt-1)}+λ-2∑i<jαiαjE(It-1,iεt-1)E(It-1,jεt-1)=∑i=1m{[piαi2(ui2+σi2)-pi2αi2ui2]+piαi(1-αi)ui}+λ-2∑i<jpiαiuipjαjuj=∑i=1mpiαi[αi(ui2+σi2)+(1-αi)ui]-(∑i=1mpiαiui)2+λ.
(iii) Note the correlation between αi∘εt-1 and εt-1; we have
(6)cov(Xt,Xt-1)=cov(∑i=1mIt-1,i(αi∘εt-1)+εt,∑j=1mIt-2,j(αj∘εt-2)+εt-1)=∑i=1mcov(It-1,i(αi∘εt-1)+εt,εt-1)=∑i=1m{E[It-1,i(αi∘εt-1)εt-1]-E[It-1,i(αi∘εt-1)]E(εt-1)}=∑i=1m{E[E(It-1,i(αi∘εt-1)εt-1∣εt-1)]vv-E[E(It-1,i(αi∘εt-1)∣εt-1)]E(εt-1)}=∑i=1mαi[E(It-1,iεt-12)-λαiE(It-1,iεt-1)]=∑i=1mpiαi(ui2+σi2-λui).
Theorem 3.
Let Xt be the process defined by the equation in (1); then the {Xt} is a covariance stationary process.
Proof.
Both the unconditional mean and the unconditional variance of the {Xt} are finite constant. And the autocovariance function does not change with time. Thus {Xt}is a stationary process.
Theorem 4.
Suppose {Xt} is INMASC(1) process. Then
T(X--μX)→LN(0,σX2+2γX(1));
E(Xtk∣It-1,i=1)<∞, k=1,2,3,i=1,…,m.
Proof.
(i) From definition and Theorem 2, we have that (X1,…,Xi) and (Xj,Xj+1,…) are independent whenever j-i>1. According to Theorem 9.1 of DasGupta [25], the process {Xt} is a stationary 1-dependent sequence. Therefore we can complete the proof.
(ii) For k=1, it follows that
(7)E(Xt)≤max{E[It-1,i(αi∘εt-1)+εt],i=1,…,m}≤max{E(αi∘εt-1)+E(εt),i=1,…,m}≤λ(αmax+1)<∞,αmax=max(α1,…,αm).
For k=2,
(8)E(Xt2)≤max{E[It-1,i(αi∘εt-1)+εt]2,i=1,…,m}=max{E[It-i,1(αi∘εt-1)]2+E(εt2)vvvvv+2E[It-i,1(αi∘εt-1)εt],i=1,…,mE[It-i,1(αi∘εt-1)]2}≤max{E[(αi∘εt-1)]2+E(εt2)vvvv+2E[(αi∘εt-1)εt],i=1,…,mE[(αi∘εt-1)]2}=max{[(λ+λ2)αi2+λαi(1-αi)]vvvv+(λ+λ2)+λ2αi,i=1,…,m}≤2(λ+λ2)αmax+0.25λ+λ2αmax<∞.
For k=3,
(9)E(Xt3)≤max{E[It-1,i(αi∘εt-1)+εt]3,i=1,…,m}=max{E[It-1,i(αi∘εt-1)]3+E(εt3)vvvv+3E[It-1,i2(αi∘εt-1)2εt]vvvv+3E[It-1,i(αi∘εt-1)εt2],i=1,…,m[It-1,i(αi∘εt-1)]3}≤max{E(αi∘εt-1)3+E(εt3)+3E[(αi∘εt-1)2εt]vvvv+3E[(αi∘εt-1)εt2],i=1,…,mE(αi∘εt-1)3+E(εt3)+3E[(αi∘εt-1)2εt]}≤max{[αi3τ1+3αi2(1-αi)τ2vvvvv+(αi-3αi2(1-αi)-αi3)λ]vvvv+τ1+3{[αi2τ2+αi(1-αi)λ]λ}vvvv+3λαiτ2,i=1,…,mαi3τ1+3αi2}≤λαmax[αmax2(τ1-1-3λ)vvvvvv+3τ2(αmax+1)+3λ+1αmax2]+τ1<∞,
where τ1:=λ3+3λ2+λ, τ2:=λ2+λ, and αmax=max(α1,…,αm). Then note that E(Xtk)<∞ implies E(Xtk∣It-1,i=1)<∞ for k=1,2,3, i=1,2,…,m.
Theorem 5.
Let {Xt} be a INMASC(1) process according to Definition 1. Let X- be the sample mean of {Xt}; then the stochastic process {Xt} is ergodic in the mean.
Proof.
Since γX(k)→0, k→∞.
From Theorem 7.1.1 in Brockwell and Davis [26], we get
(10)Var(X-T)=E(X-T-μX)2⟶0.
Then X-T converges in probability to μX. Therefore, the process {Xt} is ergodic in the mean.
Theorem 6.
Suppose {Xt} is a INMASC(1) process; then
(11)P(|γ^X(k)-γX(k)|≥ε)→P0,
where γ^X(k)∶=(1/T)∑t=1T-k(Xt+k-X-T)(Xt-X-T).
The proof of Theorem 6 is similar to Theorem 4 given in Yu et al. [27]. It is easy to verify; we skip the details.
3. Estimation of Parameters
In this paper, we consider one method, namely, moment estimation. An advantage of the method is that it is simple and often produces good results. The estimation problem of INMASC(1) parameters is complex. In fact, for the INMASC(1) processes, the conditional distribution of the Xt given εt-1 is the convolution of the distribution of the arrival process εt and one thinning operation αi∘εt-1. On the other hand, there are too many unknown parameters of the model, such as λ, αi, pi, ui, and σi2, i=1,…,m, whereas the number of moment conditions is small.
Therefore we cannot estimate all the parameters unless additional assumptions are made. Then, we assume that the number of break point m is two and assume that the value of break point τi, i=1,…,m, and the mean of innovation λ are also known. Thus, here we estimate INMASC(1) model with two break points. Under these assumptions, all the parameters λ, pi, ui, and σi2, i=1,2,3, are known. We only need to estimate the autoregressive coefficients α1, α2, and α3. Using the sample mean and sample covariance function, we can get the moment estimators via solving the following equations:
(12)γ^(0)=∑i=13piαi[αi(ui2+σi2)+(1-αi)ui]-(∑i=13piαiui)2+λγ^(1)=∑i=13piαi(ui2+σi2-λui)X-=∑i=13piαiui+λ.
If you want to estimate all parameters, you can use GMM method based on probability generating functions introduced by BräKnnäK and Hall [28]. But they found covariance matrix of estimators depends on z and the orders besides the model parameters in a highly complex way. Thus we do not use this method here. In next section, simulations are provided to give insight into the finite sample behaviour of these estimators.
4. Simulation Study
Consider the following INMASC(1) model:
(13)Xt={α1∘εt-1+εt,forεt-1≤τ1α2∘εt-1+εt,forτ1<εt-1≤τ2α3∘εt-1+εt,forτ2<εt-1,
where {εt} is a sequence of i.i.d. For fixed t, εt follows a Poisson distribution with mean λ.
The parameters values considered in this model are listed as follows:
(model A) (α1,α2,α3)=(0.1,0.1,0.1), with τ1=3, τ2=10, λ=1;
(model B) (α1,α2,α3)=(0.2,0.3,0.1), with τ1=8, τ2=17, λ=10;
(model C) (α1,α2,α3)=(0.4,0.3,0.1), with τ1=21, τ2=43, λ=50.
We use the above models to generate data and then use moment methods to estimate the parameters. We computed the empirical bias and the mean square error (MSE) based on 300 replications for each parameter combination. These values are reported within parenthesis in Table 1.
Bias and mean square error for models A, B, and C.
Model
Parameter
Sample size
50
200
500
A
α1
0.0267(0.3948)
0.0097(0.0753)
0.0034(0.0453)
BiasMSE
α2
0.0645(0.4731)
0.0115(0.1314)
0.0025(0.0376)
BiasMSE
α3
0.0417(0.2908)
0.0083(0.0811)
0.0046(0.0342)
BiasMSE
B
α1
0.0335(0.4623)
0.0127(0.1803)
0.0036(0.0745)
BiasMSE
α2
0.0297(0.2806)
0.0103(0.3449)
0.0054(0.0847)
BiasMSE
α3
0.0251(0.3408)
0.0081(0.0372)
0.0024(0.0165)
BiasMSE
C
α1
0.0736(1.0435)
0.0178(0.3562)
0.0068(0.0357)
BiasMSE
α2
0.0582(0.4127)
0.0215(0.0433)
0.0049(0.0212)
BiasMSE
α2
0.0237(0.3205)
0.0081(0.0547)
0.0031(0.0274)
BiasMSE
From the results in Table 1, we can see moment estimation is good estimation methods producing estimators whose bias and MSEs are small when the sample sizes are larger. In addition, this method is fast and easy to implement. It is perhaps not surprising that the MSEs are larger when these sample sizes are smaller. As to be expected, both the bias and the MSEs converge to zero with increasing sample size T.
5. Conclusion
Based on some limitations of the present count data models, a new INMA model is introduced to model structural changes. Expressions for mean, variance, and autocorrelation functions are given. Stationary and other basic statistical properties are also obtained. We derived moment estimators of the unknown parameters. Furthermore, we constructed several simulations to evaluate the performance of the estimators of model parameters.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 71201126), the Science Foundation of Ministry of Education of China (no. 12XJC910001), and the Fundamental Research Funds for the Central Universities (nos. JBK140211 and JBK120405).
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