Empirical Likelihood Inference for First-Order Random Coefficient Integer-Valued Autoregressive Processes

Integer-valued time series data are fairly common in practice. Especially in economics and medicine, many interesting variables are integer-valued. In the last three decades, integervalued time series have received increasing attention because of their wide applicability in many different areas, and there were many developments in the literature on it. See, for instance, Davis et al. [1] and MacDonald and Zucchini [2]. For count data, so far there are twomain classes of time series models that have been developed in recent years: state-space models and thinningmodels. For state-spacemodels, we refer to Fukasawa and Basawa [3]. Integer-valued autoregressive (INAR(1)) model was first defined by Steutel and Harn [4] through the “thinning” operator ∘. Recall the definition of a “thinning” operator ∘:


Introduction
Integer-valued time series data are fairly common in practice.Especially in economics and medicine, many interesting variables are integer-valued.In the last three decades, integervalued time series have received increasing attention because of their wide applicability in many different areas, and there were many developments in the literature on it.See, for instance, Davis et al. [1] and MacDonald and Zucchini [2].For count data, so far there are two main classes of time series models that have been developed in recent years: state-space models and thinning models.For state-space models, we refer to Fukasawa and Basawa [3].Integer-valued autoregressive (INAR(1)) model was first defined by Steutel and Harn [4] through the "thinning" operator ∘.Recall the definition of a "thinning" operator ∘: where  is an integer-valued random variable and  ∈ [0, 1] and   is an i.i.d.Bernoulli random sequence with (  = 1) =  that is independent of .Based on the "thinning" operator ∘, the INAR(1) model is defined as where {  } is a sequence of i.i.d.nonnegative integer-valued random variables.
The remainder of this paper is organized as follows: In Section 2, we introduce the methodology and the main results.Simulation results are given in Section 3. Section 4 provides the proofs of the main results.
Throughout the paper, we use the notations "  →" and "  →" to denote convergence in distribution and convergence in probability, respectively.Convergence "almost surely" is written as "a.s."Furthermore,   × denotes the transpose matrix of the  ×  matrix  × , and ‖ ⋅ ‖ denotes Euclidean norm of the matrix or vector.

Methodology and Main Results
In this section, we will first discuss how to apply the empirical likelihood method to estimate the unknown parameter . For simplicity of notation, we write   (, ) as   ; parameters  and  will be omitted.Then, after simple algebra, we get .First we consider estimating  by using the conditional least-squares method.Based on the sample  0 ,  1 , . . .,   , the least-squares estimator β of  can be obtained by minimizing with .Solving the equation for , we have Let β = β( φ, λ), where φ and λ are given by Zheng et al. [5].Further let θ = (0, 1, 1) β and  * = ( 2  , θ )  .Then, the estimating equation of  can be written as where  = (1, 0, 0)  .
In what follows, we apply Owen's empirical likelihood method to make inference about  2  .For convenience of writing, let By using the Lagrange multiplier method, introducing a Lagrange multiplier  ∈ , we have where  satisfies Owen's empirical log-likelihood ratio statistic has a chisquared limiting distribution.Similarly, we can prove that ( 2  ) will also be asymptotically chi-squared distributed.In order to establish a theory for ( 2  ), we assume that the following assumptions hold: (A 1 ) {  } is a strictly stationary and ergodic process.
Remark 1. Similar conditions can be found in [8].Now we can give the limiting properties of ( 2  ).
Theorem 2. Assume that (A 1 ) and where  2 1 is a chi-squared distribution with 1 degree of freedom.
As a consequence of the theorem, confidence regions for the parameter  2   can be constructed by (12).For 0 <  < 1, an asymptotic 100(1 − )% confidence region for  2   is given by where  2 1 () is the upper -quantile of the chi-squared distribution with degrees of freedom equal to 1.

Simulation Study
In this section, we conduct some simulation studies which show that our proposed methods perform very well.
In the first simulation study, we consider the RCINAR(1) process:  1.The nominal confidence level is chosen to be 0.90 and 0.95, and the figures in parentheses are the simulation results at the nominal level of 0.90.
From Table 1, we find that the confidence region obtained by using the empirical likelihood method has high coverage levels for different  2   .The coverage probability has no obvious change for different  and .That means that the empirical likelihood method is also robust.
In the second simulation study, we illustrate how our method can be applied to fit a set of data through a practical example.We apply model (3) to fit the number of large-and medium-sized civil Boeing 767 cargo planes over the period 1985-2013 in China.The data in Table 2 are provided by the National Bureau of Statistics of China (http://data.stats.gov.cn/easyquery.htm?cn=C01).The fitting procedure is as follows: Firstly, by using the data over the period 1985-2003, we obtain the estimator of the model parameter.Then, by using this model, we can obtain a fitting sequence over the period 2004-2013.Furthermore, in order to compare with the ordinary autoregressive (AR(1)) model, we also give the fitting results of the AR(1) model.Table 3 reports the fitting results.In Table 3, Number is the true value and RCINAR (1) and AR(1) are the fitting results obtained by the RCINAR(1) model and AR(1) model, respectively.For the simulation results of AR(1) model, we take the rounded integer values of the simulation results.From the simulation results, we can find that the RCINAR(1) model has more plausible fitting results than the AR(1) model.

Table 1 :
Coverage probabilities of the confidence intervals on  2  .{  } is a sequence of i.i.d.sequence with (  ) =  and Var(  ) =  2  ;   ∼ Poisson().We take  = 0.1, 0.2, 0.3, 0.4 and 0.5 and take  = 1 and 2. Samples of size  = 50, 100 and 300.All simulation studies are based on 1000 repetitions.The results of the simulations are presented in Table where

Table 2 :
The number of Boeing 767 cargo planes.

Table 3 :
The fitting results.