We apply the empirical likelihood method to estimate the variance of random coefficient in the first-order random coefficient integer-valued autoregressive (RCINAR(1)) processes. The empirical likelihood ratio statistic is derived and some asymptotic theory for it is presented. Furthermore, a simulation study is presented to demonstrate the performance of the proposed method.
1. 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, integer-valued 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 ∘: (1)ϕ∘X=∑i=1XBi,where X is an integer-valued random variable and ϕ∈[0,1] and Bi is an i.i.d. Bernoulli random sequence with P(Bi=1)=ϕ that is independent of X. Based on the “thinning” operator ∘, the INAR(1) model is defined as(2)Xt=ϕ∘Xt-1+Zt,t≥1,where {Zt} is a sequence of i.i.d. nonnegative integer-valued random variables.
Note that the parameter ϕ may be random and it may vary with time; Zheng et al. [5] introduced the following first-order random coefficient integer-valued autoregressive (RCINAR(1)) model:(3)Xt=ϕt∘Xt-1+Zt,t≥1,where {ϕt} is an independent identically distributed sequence with cumulative distribution function pϕ on [0,1) with E(ϕt)=ϕ and Var(ϕt)=σϕ2; {Zt} is a sequence of i.i.d. nonnegative integer-valued random variables with E(Zt)=λ and Var(Zt)=σZ2. Moreover, {ϕt} and {Zt} are independent.
Zheng et al. [6] further generalized the above model to the p-order cases. In recent several years, RCINAR model has been studied by many authors (see references in [7–10]). In this paper, we are concerned with estimating the variance σϕ2 of random coefficient in model (3). We propose an empirical log-likelihood ratio statistics for σϕ2 and derive its asymptotic distribution which is standard χ2.
As a nonparametric statistical method, the empirical likelihood method was introduced by Owen [11–13]. The advantages of the empirical likelihood are now widely recognized. It has sampling properties similar to the bootstrap. Many advantages of the empirical likelihood over the normal approximation-based method have also been shown in the literature. These attractive properties have motivated various authors to extend empirical likelihood methodology to other situations. Now, the empirical likelihood methods have been widely applied to the statistical inference of the time series models (see [14–21]).
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 “→d” and “→p” to denote convergence in distribution and convergence in probability, respectively. Convergence “almost surely” is written as “a.s.” Furthermore, Ak×pτ denotes the transpose matrix of the k×p matrix Bk×p, and · denotes Euclidean norm of the matrix or vector.
2. Methodology and Main Results
In this section, we will first discuss how to apply the empirical likelihood method to estimate the unknown parameter σϕ2.
Let θ=(ϕ(1-ϕ)-σϕ2,σZ2)τ, β=(σϕ2,θτ)τ and Rt(ϕ,λ)=Xt-E(Xt∣Xt-1). For simplicity of notation, we write Rt(ϕ,λ) as Rt; parameters ϕ and λ will be omitted. Then, after simple algebra, we get E(Xt∣Xt-1)=ϕXt-1+λ and E(Rt2∣Xt-1)=Ztτβ, where Zt=(Xt-12,Xt-1,1)τ.
First we consider estimating β by using the conditional least-squares method. Based on the sample X0,X1,…,Xn, the least-squares estimator β^ of β can be obtained by minimizing (4)Q=∑t=1nRt2-ERt2∣Xt-12with β. Solving the equation(5)∂Q∂β=∑t=1nRt2-ERt2∣Xt-1Ztfor β, we have(6)β^=∑t=1nZtZtτ-1∑t=1nRt2Zt.Let β~=β^(ϕ^,λ^), where ϕ^ and λ^ are given by Zheng et al. [5]. Further let θ~=(0,1,1)β~ and β∗=(σϕ2,θ~τ)τ. Then, the estimating equation of θ can be written as(7)∑t=1nTτ1n∑t=1nZtZtτ-1ZtRt2ϕ^,λ^-Ztτβ∗=0,where T=(1,0,0)τ.
In what follows, we apply Owen’s empirical likelihood method to make inference about σϕ2. For convenience of writing, let (8)Htσϕ2=Tτ1n∑t=1nZtZtτ-1ZtRt2ϕ^,λ^-Ztτβ∗;p=(p1,…,pn)τ be a probability vector with ∑t=1npt=1 and pt≥0; also, let σϕ02 denote the true parameter value for σϕ2. The log empirical likelihood ratio evaluated at σϕ2, a candidate value of σϕ02, is (9)lσϕ2=-2max∑t=1nptHtσϕ2=0∑t=1nlognpt. By using the Lagrange multiplier method, introducing a Lagrange multiplier λ∈R, we have(10)lσϕ2=2∑t=1nlog1+λτHtσϕ2,where λ satisfies(11)1n∑t=1nHtσϕ21+λτHtσϕ2=0.
Owen’s empirical log-likelihood ratio statistic has a chi-squared limiting distribution. Similarly, we can prove that l(σϕ2) will also be asymptotically chi-squared distributed. In order to establish a theory for l(σϕ2), we assume that the following assumptions hold:
{Xt} is a strictly stationary and ergodic process.
E|Xt|8<∞.
Remark 1.
Similar conditions can be found in [8].
Now we can give the limiting properties of l(σϕ2).
Theorem 2.
Assume that (A1) and (A2) hold. If σϕ02 is the true value of σϕ2, then(12)lσϕ02→dχ12asn⟶∞,where χ12 is a chi-squared distribution with 1 degree of freedom.
As a consequence of the theorem, confidence regions for the parameter σϕ2 can be constructed by (12). For 0<δ<1, an asymptotic 100(1-δ)% confidence region for σϕ2 is given by (13)Cσϕ2δ=σϕ2∈R:lσϕ2≤χ12δ,where χ12(δ) is the upper δ-quantile of the chi-squared distribution with degrees of freedom equal to 1.
3. 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:(14)Xt=ϕt∘Xt-1+Zt,t≥1,where {ϕt} is a sequence of i.i.d. sequence with E(ϕt)=ϕ and Var(ϕt)=σϕ2; Zt~Poisson(λ). We take ϕ=0.1,0.2,0.3,0.4 and 0.5 and take λ=1 and 2. Samples of size n=50,100 and 300. All simulation studies are based on 1000 repetitions. The results of the simulations are presented in Table 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.
Coverage probabilities of the confidence intervals on σϕ2.
ϕ
n = 50
n = 100
n = 300
λ = 1
0.1
0.980 (0.960)
0.976 (0.949)
0.979 (0.956)
0.2
0.972 (0.946)
0.984 (0.971)
0.984 (0.966)
0.3
0.977 (0.958)
0.986 (0.959)
0.990 (0.970)
0.4
0.983 (0.943)
0.984 (0.958)
0.984 (0.968)
0.5
0.978 (0.963)
0.986 (0.966)
0.990 (0.977)
λ = 2
0.1
0.980 (0.952)
0.972 (0.943)
0.970 (0.975)
0.2
0.977 (0.956)
0.980 (0.980)
0.973 (0.979)
0.3
0.989 (0.961)
0.983 (0.967)
0.969 (0.978)
0.4
0.983 (0.969)
0.983 (0.966)
0.974 (0.981)
0.5
0.973 (0.968)
0.982 (0.965)
0.970 (0.971)
From Table 1, we find that the confidence region obtained by using the empirical likelihood method has high coverage levels for different σϕ2. The coverage probability has no obvious change for different ϕ and λ. 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.
The number of Boeing 767 cargo planes.
Year
1985
1986
1987
1988
1989
1990
Number
2
2
4
5
6
6
Year
1991
1992
1993
1994
1995
1996
Number
6
10
12
16
17
17
Year
1997
1998
1999
2000
2001
2002
Number
17
15
16
16
17
18
Year
2003
2004
2005
2006
2007
2008
Number
22
27
27
29
22
22
Year
2009
2010
2011
2012
2013
Number
19
18
15
13
11
The fitting results.
Year
2004
2005
2006
2007
2008
2009
Number
27
27
29
22
22
19
RCINAR(1)
23
22
22
21
19
18
AR(1)
22
22
23
23
23
25
Year
2010
2011
2012
2013
Number
18
15
13
11
RCINAR(1)
18
18
16
16
AR(1)
25
27
27
29
4. Proofs of the Main ResultsLemma 3.
Assume that (A1) and (A2) hold. Then(15)1n∑t=1nHtσϕ2→dN0,TτΓ-1WΓ-1T,where W=E(ZtZtτ(Rt2-Ztτβ)2) and Γ=E(ZtZtτ).
First, we consider An3. After simple algebra calculation, we have(17)An3=Tτ1n∑t=1nZtZtτ-11n∑t=1nZtZtτβ-β∗=Tτ1n∑t=1nZtZtτ-11n∑t=1nZtZtτnβ-β∗=nTτβ-β∗=0.
Next, we consider An1. By the mean value theorem, we have (18)Rt2ϕ^,λ^-Rt2ϕ,λ=-2Rtϕ∗,λ∗Xt-1ϕ^-ϕ+λ^-λ,where ϕ∗ lies between ϕ^ and ϕ and λ∗ lies between λ^ and λ. Therefore,(19)An1=Tτ1n∑t=1nZtZtτ-11n∑t=1nZtRt2ϕ^,λ^-Rt2ϕ,λ=Tτ1n∑t=1nZtZtτ-1-2n∑t=1nZtRtϕ∗,λ∗Xt-1ϕ^-ϕ+λ^-λ=Tτ1n∑t=1nZtZtτ-1-2n∑t=1nZtRtϕ,λ+Rtϕ∗,λ∗-Rtϕ,λXt-1ϕ^-ϕ+λ^-λ=Tτ1n∑t=1nZtZtτ-1-2n∑t=1nZtϕ-ϕ∗Xt-1+λ-λ∗+Rtϕ,λXt-1ϕ^-ϕ+λ^-λ=Tτ1n∑t=1nZtZtτ-1-2n∑t=1nZtRtϕ,λXt-1ϕ^-ϕ-Tτ1n∑t=1nZtZtτ-12n∑t=1nZtRtϕ,λλ^-λ-Tτ1n∑t=1nZtZtτ-12n∑t=1nZtϕ-ϕ∗Xt-12ϕ^-ϕ-Tτ1n∑t=1nZtZtτ-12n∑t=1nZtλ-λ∗Xt-1ϕ^-ϕ-Tτ1n∑t=1nZtZtτ-12n∑t=1nZtϕ-ϕ∗Xt-1λ^-λ-Tτ1n∑t=1nZtZtτ-12n∑t=1nZtλ-λ∗λ^-λ≜Bn1+Bn2+Bn3+Bn4+Bn5+Bn6.
Below, we prove that Bni=op(1),i=1,2,3,4,5,6. For Bn1, note that (20)Bn1=Tτ1n∑t=1nZtZtτ-1-2n∑t=1nZtRtϕ,λXt-1nϕ^-ϕ.By Theorem 3.1 in Zheng et al. [5], we know that(21)nϕ^-ϕ=Op1.Moreover, by the ergodic theorem, we have(22)-2n∑t=1nZtRtϕ,λXt-1→p-2EZtRtϕ,λXt-1,(23)1n∑t=1nZtZtτ→pΓ.Further note that(24)EZtRtϕ,λXt-1=EERtϕ,λZtXt-1∣Ft-1=EZtXt-1ERtϕ,λ∣Ft-1=0,which, combined with (22) and (23), implies that(25)Bn1=Op1op1Op1=op1.Similarly, we can prove that(26)Bn2=op1.
Next, we prove that(27)Bn3=op1.Note that(28)Bn3≤1n∑t=1nZtZtτ2n∑t=1nZtXt-12ϕ-ϕ∗ϕ^-ϕ≤1n∑t=1nZtZtτ2n∑t=1nZtXt-12ϕ-ϕ∗2≤1n∑t=1nZtZtτ1n2n∑t=1nZtXt-12nϕ^-ϕ2.By the ergodic theorem, we have(29)2n∑t=1nZtXt-12=Op1.By (21), we have(30)nϕ^-ϕ2=Op1.Therefore, by (21), we have(31)Bn3=Op1op1Op1Op1=op1.Similarly, we can prove that(32)Bni=op1,i=4,5,6.Using this, together with (25), (26), and (27), we can prove(33)An1=op1.Finally, we prove that(34)An2→dN0,TτΓ-1WΓ-1T.For this, we first prove that(35)1n∑t=1nZtRt2-Ztτβ→dN0,W.By the Cramer-Wold device, it suffices to show that, for all c∈R3∖(0,0,0),(36)1n∑t=1ncτZtRt2-Ztτβ→dN0,cτWc.Let ξnt=(1/n)cτZt(Rt2-Ztτβ) and Fnt=σ(ξnr,1≤r≤t). Then {∑t=1nξnt,Fnt,1≤t≤n,n≥1} is a zero-mean, square integrable martingale array. By making use of a martingale central limit theorem [22], we can prove (36). Further, by (23), we know that (34) holds. Therefore, by (17), (33), and (34), we can prove Lemma 3.
Lemma 4.
Assume that (A1) and (A2) hold. Then(37)1n∑t=1nHt2σϕ2→pTτΓ-1WΓ-1T.
Proof.
Note that(38)1n∑t=1nHt2σϕ2=Tτ1n∑t=1nZtZtτ-11n∑t=1nZtZtτRt2ϕ^,λ^-Ztτβ∗21n∑t=1nZtZtτ-1T.By (23), in order to prove Lemma 4, we have only to show that(39)1n∑t=1nZtZtτRt2ϕ^,λ^-Ztτβ∗2→pW.Note that(40)1n∑t=1nZtZtτRt2ϕ^,λ^-Ztτβ∗2=1n∑t=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λ+Rt2ϕ,λ-Ztτβ+Ztτβ-Ztτβ∗2=1n∑t=1nZtZtτRt2ϕ,λ-Ztτβ2+1n∑t=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λ2+1n∑t=1nZtZtτZtτβ-Ztτβ∗2+2n∑t=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λRt2ϕ,λ-Ztτβ+2n∑t=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λZtτβ-Ztτβ∗+2n∑t=1nZtZtτRt2ϕ,λ-ZtτβZtτβ-Ztτβ∗≜Cn1+Cn2+Cn3+Cn4+Cn5+Cn6.By the ergodic theorem, we know that(41)Cn1→pW.Similar to the proof of (33), we can further prove that(42)Cni=op1,i=2,3,4,5,6.This, in conjunction with (41), yields (39). So we complete the proof of Lemma 4.
Lemma 5.
Assume that (A1) and (A2) hold. Then(43)max1≤t≤nHtσϕ2=opn1/2.
Proof.
To prove (43), we only need to prove that(44)1nmax1≤t≤nHt2σϕ2→p0.Let TτΓ-1WΓ-1T=σ2. For m∈{1,…,n}, define(45)Bn,m≜⋂j=1nω:1n∑t=1nj/mHtσϕ2-jmσ2≤1m,where [n(j/m)] denotes the largest integer not greater than n(j/m). For each m, (37) implies that P(Bn,m)→P1 as n→∞. Moreover, note that (46)max1≤t≤nHtσϕ2n≤1nsups∈0,1∑t=ns+1ns+1/mHtσϕ2. For given s∈[0,1], choose j∈{1,…,m} so that s∈[(j-1)/m,j/m]. Therefore, for each s∈[0,1], if ω∈Bn,m, then we have (47)1n∑t=ns+1ns+1/mHtσϕ2≤1n∑t=nj-1/m+1nj+1/mHtσϕ2=1n∑t=1nj+1/mHtσϕ2-j+1mσ2-1n∑t=1nj-1/mHtσϕ2-j-1mσ2+2mσ2≤2m+2mσ2=2m1+σ2. So, for any m≥1,(48)limn→∞Pmax1≤t≤nHtσϕ2n≤2m1+σ2≥limn→∞PBn,m=1,which implies (44). So we prove (43).
Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref>.
First, we prove that(49)λ=Opn-1/2.Write λ=ρϑ, where ρ≥0 and |ϑ|=1. Observe that (50)0=1n∑t=1nHtσϕ21+λHtσϕ2=ϑ1n∑t=1nHtσϕ21+λHtσϕ2=1nϑ∑t=1nHtσϕ2-ρ∑t=1nϑHt2σϕ21+ρϑHtσϕ2≥ρnϑ2∑t=1nHt2σϕ21+ρϑHtσϕ2-1nϑ∑t=1nHtσϕ2≥ρnϑ2∑t=1nHt2σϕ21+ρmax1≤t≤nHtσϕ2-1nϑ∑t=1nHtσϕ2≥11+ρmax1≤t≤nHtσϕ2ρϑ21n∑t=1nHt2σϕ2-1nϑ∑t=1nHtσϕ2.This implies that(51)11+ρmax1≤t≤nHtσϕ2ρϑ21n∑t=1nHt2σϕ2≤1nϑ∑t=1nHtσϕ2.Further, by Lemma 4, we know that (52)ϑ21n∑t=1nHt2σϕ2=Op1.By Lemma 3, we have (53)1nϑ∑t=1nHtσϕ2=Opn-1/2.Thus by (51) and Lemma 5, we have (54)ρ=λ=Opn-1/2,which implies (49).
By (49) and Lemma 5, we can prove that(55)max1≤t≤nλHtσϕ2=Opn-1/2opn1/2=op1.Expanding (11), we have(56)0=1n∑t=1nHtσϕ21+λHtσϕ2=1n∑t=1nHtσϕ2-1nλ∑t=1nHt2σϕ2+1n∑t=1nλ2Ht3σϕ21+λHtσϕ2.By (55) and Lemmas 3, 4, and 5, we know that the final term in (56) is bounded by(57)1n∑t=1nHtσϕ23λ21+λHtσϕ2-1=opn1/2Opn-1Op1=opn-1/2.This, together with (41), yields(58)λ=1n∑t=1nHt2σϕ2-11n∑t=1nHtσϕ2+opn-1/2.By the Taylor expansion, we have (59)log1+λHtσϕ2=λHtσϕ2-λHtσϕ222+φt.Below, we prove that there exists a finite number Q>0, such that(60)Pφt≤QλHtσϕ23,1≤t≤n⟶1asn⟶∞.The Taylor expansion of log(1+x) around x=0 yields (61)log1+x=x-x22+x33+ϖx,where, as x→0,ϖ(x)/x3→0. Thus, there exists ι>0, such that ϖ(x)/x3<1/6 for any |x|<ι. Moreover, by (55), we have (62)limn→∞Pmax1≤t≤nλHtσϕ23<ι3=1.Let An={ω:max1≤t≤n|λHt(σϕ2)|3<ι3}. Note that if ω∈An, then for 1≤t≤n,(63)φtλHtσϕ23=λHtσϕ23/3+ϖλHtσϕ2λHtσϕ23≤13+16=12,which implies that (64)Pφt≤QλHtσϕ23,1≤t≤n⟶1asn⟶∞,where Q=1/2.
Moreover, by (10) and (58), we have(65)lσϕ2=1n∑t=1nHtσϕ21n∑t=1nHt2σϕ2-11n∑t=1nHtσϕ2+op1.This, together with Lemmas 3 and 4, implies Theorem 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors acknowledge the financial supports by National Natural Science Foundation of China (nos. 11571138, 11271155, 11001105, 11071126, 10926156, and 11071269), Specialized Research Fund for the Doctoral Program of Higher Education (no. 20110061110003), Program for New Century Excellent Talents in University (NCET-08-237), Scientific Research Fund of Jilin University (no. 201100011), and Jilin Province Natural Science Foundation (nos. 20130101066JC, 20130522102JH, and 20101596).
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