Integer-Valued Moving Average Models with Structural Changes

Abstract

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 Properties

Definition 1. Let {𝑋_𝑡} be a process with state space ℕ₀; let 0<𝛼_𝑖<1, 𝑖=1,…,𝑚, and 𝜏_𝑖, 𝑖=1,…,𝑚−1, be positive integers. The process {𝑋_𝑡} is said to be first order integer-valued moving average model with structural change (INMASC(1)) if 𝑋_𝑡 satisfies the following equation: 𝑋_𝑡=𝛼₁∘𝜀_𝑡−1+𝜀_𝑡,for𝜏₀≤𝜀_𝑡−1≤𝜏₁𝛼₂∘𝜀_𝑡−1+𝜀_𝑡,for𝜏₁<𝜀_𝑡−1≤𝜏₂⋮𝛼_𝑚−1∘𝜀_𝑡−1+𝜀_𝑡,for𝜏_𝑚−2<𝜀_𝑡−1≤𝜏_𝑚−1𝛼_𝑚∘𝜀_𝑡−1+𝜀_𝑡,for𝜏_𝑚−1<𝜀_𝑡−1<𝜏_𝑚, (1) where {𝜀_𝑡} is a sequence of independent and identically distributed Poisson random variables with mean 𝜆 and 𝜏₀∶=0, 𝜏_𝑚∶=∞.

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: 𝑝_𝑖∶=𝑃𝜏_𝑖−1<𝜀_𝑡≤𝜏_𝑖,𝑢_𝑖∶=𝐸𝜏_𝑖−1<𝜀_𝑡−1≤𝜏_𝑖,𝜎²_𝑖∶=Var𝜏_𝑖−1<𝜀_𝑡−1≤𝜏_𝑖,𝑞_𝑖∶=1−𝑝_𝑖,𝐼_{𝑡−1,𝑖}∶=1,if𝜏_𝑖−1<𝜀_𝑡−1≤𝜏_𝑖0,otherwise,𝑖=1,…,𝑚. (2)

Theorem 2. The numerical characteristics of {𝑋_𝑡} are as follows: (𝑖)𝜇_𝑋∶=𝐸𝑋_𝑡=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖()𝜎+𝜆,𝑖𝑖²_𝑋∶=Var𝑋_𝑡=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖+1−𝛼_𝑖𝑢_𝑖−^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖²()𝛾+𝜆,𝑖𝑖𝑖_𝑋(𝑘)∶=cov𝑋_𝑡,𝑋_𝑡−𝑘=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖−𝜆𝑢_𝑖,𝑘=10,𝑘≥2. (3)

Proof.  (i) It is easy to get the mean and variance of 𝑋_𝑡 by using the law of iterated expectations: 𝐸𝑋_𝑡=𝐸𝐼_𝑡−1,1𝛼₁∘𝜀_𝑡−1+⋯+𝐼_{𝑡−1,𝑚}𝛼_𝑚∘𝜀_𝑡−1+𝜀_𝑡=𝐸𝐸𝐼_𝑡−1,1𝛼₁∘𝜀_𝑡−1+⋯+𝐼_{𝑡−1,𝑚}𝛼_𝑚∘𝜀_𝑡−1∣𝜀_𝑡−1+𝐸𝜀_𝑡=𝛼₁𝐸𝐼_𝑡−1,1𝜀_𝑡−1+⋯+𝛼_𝑚𝐸𝐼_{𝑡−1,𝑚}𝜀_𝑡−1+𝐸𝜀_𝑡=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖+𝜆. (4)
(ii) Moreover, Var𝑋_𝑡=Var𝐼_𝑡−1,1𝛼₁∘𝜀_𝑡−1+⋯+𝐼_{𝑡−1,𝑚}𝛼_𝑚∘𝜀_𝑡−1+𝜀_𝑡=^𝑚_𝑖=1Var𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1+Var𝜀_𝑡+2_𝑖<𝑗cov𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1,𝐼_{𝑡−1,𝑗}𝛼_𝑗∘𝜀_𝑡−1=^𝑚_𝑖=1Var𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1∣𝜀_𝑡−1+𝐸Var𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1∣𝜀_𝑡−1+𝜆+2_𝑖<𝑗𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝐼_{𝑡−1,𝑗}𝛼_𝑗∘𝜀_𝑡−1−2𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝐸𝐼_{𝑡−1,𝑗}𝛼_𝑗∘𝜀_𝑡−1=^𝑚_𝑖=1𝛼²_𝑖Var𝐼_{𝑡−1,𝑖}𝜀_𝑡−1+𝛼_𝑖1−𝛼_𝑖𝐸𝐼_{𝑡−1,𝑖}𝜀_𝑡−1+𝜆−2_𝑖<𝑗𝐸𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1∣𝜀_𝑡−1×𝐸𝐸𝐼_{𝑡−1,𝑗}𝛼_𝑗∘𝜀_𝑡−1∣𝜀_𝑡−1=^𝑚_𝑖=1𝛼²_𝑖𝐸𝐼²_𝑡−1𝜀²_𝑡−1−𝐸²𝐼_{𝑡−1,𝑖}𝜀_𝑡−1+𝑝_𝑖𝛼_𝑖1−𝛼_𝑖𝑢_𝑖+𝜆−2_𝑖<𝑗𝛼_𝑖𝛼_𝑗𝐸𝐼_{𝑡−1,𝑖}𝜀_𝑡−1𝐸𝐼_{𝑡−1,𝑗}𝜀_𝑡−1=^𝑚_𝑖=1𝑝_𝑖𝛼²_𝑖𝑢²_𝑖+𝜎²_𝑖−𝑝²_𝑖𝛼²_𝑖𝑢²_𝑖+𝑝_𝑖𝛼_𝑖1−𝛼_𝑖𝑢_𝑖+𝜆−2_𝑖<𝑗𝑝_𝑖𝛼_𝑖𝑢_𝑖𝑝_𝑗𝛼_𝑗𝑢_𝑗=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖+1−𝛼_𝑖𝑢_𝑖−^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖²+𝜆. (5)
(iii) Note the correlation between 𝛼_𝑖∘𝜀_𝑡−1 and 𝜀_𝑡−1; we have cov𝑋_𝑡,𝑋_𝑡−1=cov^𝑚_𝑖=1𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1+𝜀_𝑡,^𝑚_𝑗=1𝐼_{𝑡−2,𝑗}𝛼_𝑗∘𝜀_𝑡−2+𝜀_𝑡−1=^𝑚_𝑖=1cov𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1+𝜀_𝑡,𝜀_𝑡−1=^𝑚_𝑖=1𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝜀_𝑡−1−𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝐸𝜀_𝑡−1=^𝑚_𝑖=1𝐸𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝜀_𝑡−1∣𝜀_𝑡−1−𝐸𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1∣𝜀_𝑡−1𝐸𝜀_𝑡−1=^𝑚_𝑖=1𝛼_𝑖𝐸𝐼_{𝑡−1,𝑖}𝜀²_𝑡−1−𝜆𝛼_𝑖𝐸𝐼_{𝑡−1,𝑖}𝜀_𝑡−1=^𝑚_𝑖=1𝑝_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖−𝜆𝑢_𝑖. (6)

Theorem 3. Let 𝑋_𝑡 be the process defined by the equation in (1); then the {𝑋_𝑡} is a covariance stationary process.

Proof. Both the unconditional mean and the unconditional variance of the {𝑋_𝑡} are finite constant. And the autocovariance function does not change with time. Thus {𝑋_𝑡}is a stationary process.

Theorem 4. Suppose {𝑋_𝑡} is INMASC(1) process. Then(i)√𝑇(𝑋−𝜇_𝑋)→−^𝐿𝑁(0,𝜎²_𝑋+2𝛾_𝑋(1));(ii)𝐸(𝑋^𝑘_𝑡∣𝐼_{𝑡−1,𝑖}=1)<∞, 𝑘=1,2,3,𝑖=1,…,𝑚.

Proof. (i) From definition and Theorem 2, we have that (𝑋₁,…,𝑋_𝑖) and (𝑋_𝑗,𝑋_𝑗+1,…) are independent whenever 𝑗−𝑖>1. According to Theorem 9.1 of DasGupta [25], the process {𝑋_𝑡} is a stationary 1-dependent sequence. Therefore we can complete the proof.
(ii) For 𝑘=1, it follows that 𝐸𝑋_𝑡≤max𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1+𝜀_𝑡𝐸𝛼,𝑖=1,…,𝑚≤max_𝑖∘𝜀_𝑡−1+𝐸𝜀_𝑡𝛼,𝑖=1,…,𝑚≤𝜆_max+1<∞,𝛼_max=max𝛼₁,…,𝛼_𝑚. (7) For 𝑘=2, 𝐸𝑋²_𝑡≤max𝐸[𝐼_{𝑡−1,𝑖}(𝛼_𝑖∘𝜀_𝑡−1)+𝜀_𝑡]²𝐸𝐼,𝑖=1,…,𝑚=max_{𝑡−𝑖,1}𝛼_𝑖∘𝜀_𝑡−1²+𝐸𝜀²_𝑡+2𝐸𝐼_{𝑡−𝑖,1}𝛼_𝑖∘𝜀_𝑡−1𝜀_𝑡𝐸𝛼,𝑖=1,…,𝑚≤max_𝑖∘𝜀_𝑡−1²+𝐸𝜀²_𝑡+2𝐸𝛼_𝑖∘𝜀_𝑡−1𝜀_𝑡,𝑖=1,…,𝑚=max𝜆+𝜆²𝛼²_𝑖+𝜆𝛼_𝑖1−𝛼_𝑖+𝜆+𝜆²+𝜆²𝛼_𝑖,𝑖=1,…,𝑚≤2𝜆+𝜆²𝛼_max+0.25𝜆+𝜆²𝛼_max<∞. (8) For 𝑘=3, 𝐸𝑋³_𝑡≤max𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1+𝜀_𝑡³𝐸𝐼,𝑖=1,…,𝑚=max_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1³+𝐸𝜀³_𝑡+3𝐸𝐼²_{𝑡−1,𝑖}(𝛼_𝑖∘𝜀_𝑡−1)²𝜀_𝑡+3𝐸𝐼_{𝑡−1,𝑖}𝛼_𝑖∘𝜀_𝑡−1𝜀²_𝑡𝐸𝛼,𝑖=1,…,𝑚≤max_𝑖∘𝜀_𝑡−1³+𝐸𝜀³_𝑡+3𝐸𝛼_𝑖∘𝜀_𝑡−1²𝜀_𝑡+3𝐸𝛼_𝑖∘𝜀_𝑡−1𝜀²_𝑡𝛼,𝑖=1,…,𝑚≤max³_𝑖𝜏₁+3𝛼²_𝑖1−𝛼_𝑖𝜏₂+𝛼_𝑖−3𝛼²_𝑖1−𝛼_𝑖−𝛼³_𝑖𝜆+𝜏₁+3𝛼²_𝑖𝜏₂+𝛼_𝑖1−𝛼_𝑖𝜆𝜆+3𝜆𝛼_𝑖𝜏₂,𝑖=1,…,𝑚≤𝜆𝛼_max𝛼²_max𝜏₁−1−3𝜆+3𝜏₂𝛼_max+1+3𝜆+1+𝜏₁<∞, (9) where 𝜏₁∶=𝜆³+3𝜆²+𝜆, 𝜏₂∶=𝜆²+𝜆, and 𝛼_max=max(𝛼₁,…,𝛼_𝑚). Then note that 𝐸(𝑋^𝑘_𝑡)<∞ implies 𝐸(𝑋^𝑘_𝑡∣𝐼_{𝑡−1,𝑖}=1)<∞ for 𝑘=1,2,3, 𝑖=1,2,…,𝑚.

Theorem 5. Let {𝑋_𝑡} be a INMASC(1) process according to Definition 1. Let 𝑋 be the sample mean of {𝑋_𝑡}; then the stochastic process {𝑋_𝑡} is ergodic in the mean.

Proof. Since 𝛾_𝑋(𝑘)→0, 𝑘→∞.
From Theorem 7.1.1 in Brockwell and Davis [26], we get Var𝑋_𝑇=𝐸𝑋_𝑇−𝜇_𝑋²⟶0. (10) Then 𝑋_𝑇 converges in probability to 𝜇_𝑋. Therefore, the process {𝑋_𝑡} is ergodic in the mean.

Theorem 6. Suppose {𝑋_𝑡} is a INMASC(1) process; then 𝑃𝛾_𝑋(𝑘)−𝛾_𝑋(𝑘)≥𝜀−→^𝑃0, (11) where 𝛾_𝑋(𝑘)∶=(1/𝑇)∑^𝑇−𝑘_𝑡=1(𝑋_𝑡+𝑘−𝑋_𝑇)(𝑋_𝑡−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 𝑋_𝑡 given 𝜀_𝑡−1 is the convolution of the distribution of the arrival process 𝜀_𝑡 and one thinning operation 𝛼_𝑖∘𝜀_𝑡−1. On the other hand, there are too many unknown parameters of the model, such as 𝜆, 𝛼_𝑖, 𝑝_𝑖, 𝑢_𝑖, and 𝜎²_𝑖, 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 𝜏_𝑖, 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, 𝑝_𝑖, 𝑢_𝑖, and 𝜎²_𝑖, 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: 𝛾(0)=³_𝑖=1𝑝_𝑖𝛼_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖+1−𝛼_𝑖𝑢_𝑖−³_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖²𝛾(1)=+𝜆_3𝑖=1𝑝_𝑖𝛼_𝑖𝑢²_𝑖+𝜎²_𝑖−𝜆𝑢_𝑖𝑋=³_𝑖=1𝑝_𝑖𝛼_𝑖𝑢_𝑖+𝜆. (12) 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: 𝑋_𝑡=𝛼₁∘𝜀_𝑡−1+𝜀_𝑡,for𝜀_𝑡−1≤𝜏₁𝛼₂∘𝜀_𝑡−1+𝜀_𝑡,for𝜏₁<𝜀_𝑡−1≤𝜏₂𝛼₃∘𝜀_𝑡−1+𝜀_𝑡,for𝜏₂<𝜀_𝑡−1, (13) where {𝜀_𝑡} is a sequence of i.i.d. For fixed 𝑡, 𝜀_𝑡 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.

Table 1 

Bias and mean square error for models A, B, and C.

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).