Some Results Of Empirical Likelihood Inference For Nonlinear Time Series Models

In this doctoral dissertation, three nonlinear time series models are investigat-ed: the partly linear autoregressive model, the functional coefficient model with the dependent samples and the autoregressive conditional duration model.Consider the partly linear autoregressive model yt=Yt-1τβ+g(yt-p)+et,(1) where β is a p-1unknown parameter vector, g(?) is a unknown function, et is a stochastic error with Eet=0and D(et)=σ2, p is the order of the modelFirstly, the empirical likelihood method is used to estimate the parameters of model (1). Assume that {yt, t=p+1,...,T} are sampled from model (1) and et=yt-Ytτβ+g(yt-p), where Yt-1=(yt-1,...,yt-P+1)τ, and in addition,{y1,,y2,...,yP} are known.Under the condition of yt-p, we have g(yt-p)=E[(yt-(βTYt-1)[yt-p]=E[yt|yt-p]-βτE[Yt-1|yt-p]=91(yt-P)-βτg2(yt-p),(2) where g1(yt-p)=E[yt|yt-p],g2(yt-p)=[921(yt-P),...,g2(p-1)(yt-p)]τ=[E(yt-i|yt-p),...,E(yt-P+1|yt-p)]τ Replacing g(yt-p) in (1) with (2), we have et(β)=yt-g1(yt-p)-βτ(Yt-1-g2(yt-p)).(3)If g1(yt-p) and g2(yt-p) are known, the empirical log-likelihood ratio function could be defined by where n=T-p, ut-p=Yt-1-g2(yt-p). g1(yt-p) and g2(yt-p) are unknown, it is natural to apply the weight function method to estimate g1(yt-P), g2i(yt-P),i=1,...;p-1and g(yt-p), whose estimators are defined by and gT(yt-p)(?)g1T(yt-p)-βτg2T(yt-p), where g2T(yt-p)=[g21T(yt-p),…,g2(p-1)T(yt-p)]τ and WTs(?) are kernel weight func-tions: where K(?) is a real-valued kernel function satisfying the conditions in the appendix and h is a bandwidth satisfying that h=hT∈HT=[a1T-1/5-c1,b1T-1/2+c1], with0<a1<b1<∞,0<c1<1/20.Replacing gi(yt_p),i=1,2in (3) with (4) and (5), we obtain et-(β)=yt-g1T(yt-p)-βτ(Yt-1-g2T(yt-p))=yt-g1T(yt-p)-βτut-p,(8) where ut-p=Yt-1-g2T(yt-P).Replacing et(β) and ut-p in LT(β) with et(β) and ut-p respectively, we get By the Lagrange multiplier method, we obtain Lt(β) can be expressed as where λ=λ(β) is determined byThe following conditions are needed to obtain the asymptotic distribution of the empirical likelihood ratio for β:Assumption1.1(ⅰ) Assume that the process {yt;t≥1} is strictly stationary and α-mixing with mixing coefficient α(T)=CρT, where0<C <∞and0<ρ<1;(ⅱ) K(?) is symmetric, Lipschitz continuous, has an absolutely integrable Fourier transform and satisfies(ⅲ) K(?) is a bounded probability kernel function(ⅳ) The weight function W(?) is bounded with compact support S;(ⅴ) yt has a marginal density f(?), which has a compact support containing S, and g1(?),g2i{?),i=1,2,...,p-1, f(?) have two continuous derivatives on the interior of S.Assumption1.2For any integer k≥1, E|yt|k <∞Assumption1.3(ⅰ)There exist0<σ2<oo, such that E(et2|(?)t-1)=σ2+ο(1), a.s;(ⅱ)There exist δ≥4, such that supt>p+1E(|et|δ|(?)t-1)<∞, a.s where (?)t σ{y1,...,yp,ep+1,...,et-1}.The following theorem gives the asymptotic distribution of the empirical log-likelihood statistics for (3.Theorem1Under the Assumption1.1-1.3,βq are the true parameters, we haveIn general, p is not large in the practical problems, In the following analysis, let p be a finite known number, we denote a subset s with size k from {1,2,...,p-1} as a subset of indices of covariate variables and use et[s]=yt-g1t(yt-P)-βτ[s]ut-p[s], where β[s] is the corresponding subset of the coefficients, ut-p[s]=Yt-1[s]-g2T(yt-p)[s]. Let a subset s0be the true parameter set, it implies β0[s0]=0. Remark1Under the assumptions of Theorem1, β[s0]=0for a size k0true subset s0, we have where Lt(β0[s0]) is obtained by replacing ut_pet in LT(β) with ut-p[s0]et[s0].In the following, we will define the empirical BIC and empirical AIC based on the empirical likelihood for model (1). Let s be a subset of size k from {1,2,...,p-1}. Motivated by Variyath.et al(2010), the profile empirical log-likelihood ratio is defined by LT(s)=sup{LT(β[s]):β[s]}. The definitions are presented as follows:Definition1the empirical likelihood Akaike information criterion EAIC(s)=LT(s)+2k,Definition2the empirical likelihood Bayes information criterion EBIC(s)=LT(s)+klogT.When p is a fixed known number, we use EAIC and EBIC to select the main variables for model (1). The empirical likelihood information criteria is constructed to avoid the error hypothesis. The following theorems show that EBIC is selection consistent while EIAC is not.Theorem2Under the assumptions in Theorem1,β0[s0]=0for a size k0true subset s0, we have LT(s)→Dχp2-k-1,as T→∞When the null hypothesis of β[s]=0is not true, the empirical likelihood ratio go to oo. we have the following theoremTheorem3Under the assumptions in Theorem1, for any β≠β0, we have LT(3)→P∞, as T→∞.Let s0be the true index subset of the covariate, for any index subset s, we have for some β[s] if and only if s=s0. Then, EBIC is consistent while EAIC is not consistent.Consider the functional coefficient model for the nonlinear time series Y=α(U)τX+ε,(11) where e is a random error,(Y, Xτ, U) has the same distribution as the strictly stationary processes (Yt, Xtτ, Ut)t=-∞∞,α(?) is an unknown coefficient function vector from Rk to R. If Xj, Ui consist of some lagged values of Yi, the model (11) is the pure time series model.Let {Yi, Xiτ, Ui}in=1be sampled from {Yi, Xiτ,Ui}i=∞∞, we estimate the function α(u) and its derivative using local linear regression procedure from(11). Suppose that α{u) exists the continuous second derivative in this paper. We approximate α(u) by linear function αi(Ui)=αi(u0)+αi'(u0)(Ui-u0)=ai+bi(Ui-u0), i=1,...,p,(12) in a small neighborhood of a point u0. By minimizing the sum of weighted squares where Kh(?)=h-1K(?/h), K(?) is a kernel function on R, h>0is a bandwidth. The local linear estimators β=(aτ,bτ)τ of α(u) and α(u)' are obtained and satisfy the normal equations (XτWX)β=XτWy,(13) where β=(aT, bT)T, The following equivalent normal equations are obtained from (13) whereEquation (14) only includes the parameters a, equation (15) only includes the parameters b. The empirical likelihood ratio for the coefficient function α(?) and its derivatives are constructed based on the equations (14) and (15). LetThe auxiliary random vectors are defined as Wai(u0)=Wai(u0)(Yi-aτXi)Xi,(16) Wbi(u0)=Wbi(u0)(Yi-bτ(Ui-u0)Xi)Xi.(17) Similar to Owen(1990), the empirical log-likelihood ratio function for the coefficient function and the derivative of the coefficient function are defined by where a=α(?) and b=α'(?) are respectively the true at the point u0.By the Lagrange multiplier method, we can obtain and lan(a) can be expressed as where λa is the solution of lbn(b) can be expressed as where λb is the solution ofThe following conditions are needed to obtain some results of the empirical likeli-hood ratio for the function coefficient.Assumption2.1.(ⅰ) The process {Yt, Xiτ, Ui} is strictly stationary α-mixing with∑kc[α{k)]l-2/δ <∞for some δ>4and c>1-2/δ;(ⅱ) The kernel function K(?)is a bounded density with a bounded support[-1,1];(ⅲ) E|X|2δ<∞, where5is given in assumption2.1(i);(ⅳ)|f(u,v\x0,xl;l)|≤M <∞, for all l≤1, where f(u,v,|xo,xl;l) is the con-ditional density of U0,Ul given (X0,X1), and f(u|x)<M <∞, where f(u|x) is the conditional density of U given X=x.Assumption2.2(ⅰ) Assume that E{Y02+Yl2|X0=x0, U0=u;Xl=xl, Ul=u}≤M <∞, for all l≥1, x0, x1∈Rp, u and v in a neighborhood of u0;(ⅱ) Assume that hn—>0and nhn—>∞. Further, assume that there exists a se-quence of positive integers sn such that sn→∞, sn=op ((nhn)1/2),(nhn)1/2α(sn)→0, as n→∞;(ⅲ)There exists δ*>δ, where S is given in assumption2.1(i) such that E{|Y|δ*|X=x, U=u}≤M <∞, for all x∈Rp, and u in a neighborhood of u0, and α(n)=O(n-θ*) where θ*≥δδ*/{2(δ-δ*)};(ⅳ) E|X|2δ*<∞, and n1/2-δ/4hnδ/δ8-1=O(1).Firstly, we discuss the empirical likelihood for the derivatives of functional coeffi-cient of the varying coefficient model.Theorem4Let Assumption2.1and2.2hold,α(?) be continuous at the point u0. Let nh3→∞; nhn5→0as n→∞. α'(u0) are the true value of α'(?) at the point u=u0, it holds that lbn(α'(u0))→Dωb,1χ112+...+ωb,pχ1p2,(18) where ωb,i,1≤i≤p; are the eigenvalues of Db(u0)=∑b1/2(u0)∑b*(u0)-1∑b1/2(u0), where∑b(u0) is defined in Lemma3.5.1,∑b*(u0)is defined in Lemma3.5.2, χ1i2,1≤i≤p, are independent χ12distribution.We need to estimate the unknown weights ωi consistently in order to construct a empirical likelihood confidence region of α'(u0) based on Theorem4. From the definition of∑b(u0and∑b(u0)*, the sticking point is to estimate Ω1(y0)=E[XXτ(aτX)2|U=u0], and Ω*(u0)=E[XXτσ2(X,U)|U=u0]. From the proof of Lemma3.5.2, we estimate Ω1and Ω*where a=(α(u0),…, α(u0)) is the weight least squares estimator of (13). This implies that eigenvalues of D(u0)=∑(u0)1/2∑*(u0)-1∑(u0)1/2, ωi,1≤i≤p] say, estimate consistently for1<i <p.Let F(?) be the distribution of the weight sum ωb,1χ112+…+ωb,pχ1p2given the data {Yi,Xi,Ui}i=1n. In practice, F(?) can be obtained using Monte Carlo method by repeatedly generating independent samples χ112,…,χ1p2from the χ12distribution.Note that Monte Carlo method is needed to obtain the asymptotic distribution of the empirical log-likelihood ratio. It is computationally intensive method and the accuracy of this asymptotic distribution depends on the values of the ωi and it is difficult and fussy to obtain the the α-quantile of F(?). We recommend the adjustments to avoid the Monte Carlo simulation of the asymptotic distribution.Next, we give an adjusted empirical log-likelihood whose asymptotic distribution is exactly a standard chi-squared with p degrees of freedom.Let ρb(α(u0))=p/tr[Db(u0)], where tr[?] is the trace operator. Using Rao and Scott(1981), the distribution of ρb(α(u0))∑i=1pωiχ1i2is a standard chi-square distribu-tion with p degrees of freedom χp2. This implies that the asymptotic distribution of the Rao-Scott adjusted empirical log-likelihood ratio ρ(α(uo))ln(α'(u0)) can be approx-imated by χp2, This can be achieved by using Theorem1, the consistency of Ω*and Ω1The Rao-Scott adjusted empirical log-likelihood can be obtained by replacing α(u0) in ρ(α(u0)) by α(u0) lab(b)=ρb(α(uo))lbn(b). We have the following results.Theorem5Let Assumption2.1and2.2hold, α(?) be continuous at the point u0. Let hn→∞; nhn3→∞; nhn5→0as n→∞. α'(u0) are the true values of α'(?) at the point u=u0, lab(α'(u0))→Dχp2.Based on (18), the empirical likelihood reference for the coefficient function of Model (11) is obtained.Theorem6Let Assumption2.1and2.2hold. Let hn→0, nhn3→0, nhhn→0as n→∞.α(u0) are the true value of α(?) at the point u0, it holds that lan(α(u))→DχP2.The confident regions of α(u0) are constructed by Theorem6. There doesn't exist the problem of Theorem4in constructing the confident regions of α(u0). So, this may directly use Theorem6to construct the confident regions.Finally, consider the autoregressive conditional duration model, introduced by Enlle and Russell(1998), xt=ψtεt, εt～i.i.d,(19) where xt is the financial duration removing the intraday effect; εt are iid; ψt is the conditional duration. Because this model explains the interval of financial data, the duration xt are positive, and it is a natural assumption that ψt≥0and εt≤0, the parameters of (20) need to satisfy that ω>0, ai≥0i=1,2,...,p, βj≥0, j=1,2,...,q. The structure of models (19) and (20) is similar to that of GARCH models. Following the notations of Ling (2007), we denote Under Assumption3.1, we have∑j=1qβj<1which is equivalent to where Ik is the k identity matrix, ρ(G) is the spectral radius of matrix G, B is back-shifting operator. Under Assumption3.1, we have where supΘβ αβ{i)=O(ρi) and supΘδ αδ(i)=O(ρi). The necessary and sufficient condition for Ee2<oo isLet θ0∈Θ be the true parameters. Assume that θ0is an interior point and for any θ∈Θ, it satisfies thatAssumption3.1. ω>0,αi>0, β≥0,∑ipαi+∑iqβ<1and∑ipαiZi+∑iqβiZi have no common roots.This condition ensures the stationarity, invertibility and identifiability of models (19) and (20), and is generalized from Engle and Russell(1998).Let x1,…,xn be the observations and x0*={xt,t≤0} be the initial value. Based on Engle and Russell(1998), the quasi-likelihood function can be written as where ψt(θ) is a function of xt, θ=(ατ, βτ)τ is the parameter vector in (19) and (20). The score function is The information matrix is Let The QMLE is the solution of the score function∑t=1n Dt(θ). Using this score function Dt(θ), the log empirical likelihood function can be constructed as where λ(θ) is the Lagrange multiplier such that The global minimizer θn of (21) is the QMLE and λ(θn)=0. The empirical likelihood ratio statistics is defined for testing Ho:θ=θ0WE(θ0)=-2[LE(θn)-LE(θ0)]. We have the following result.Theorem7Assumption3.1holds, E∈t2<∞; under H0, we have WE(θO)→Dχp+q+12,asn→∞, where θn∈Vn={θ,|θ-θ0|≤n-0.5M} for each constant M>0.