# Empirical Likelihood for Partial Parameters in ARMA Models with Infinite Variance

Authors: Jinyu Li; Wei Liang; Shuyuan He
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

This paper proposes a profile empirical likelihood for the partial parameters in ARMA(p,q) models with infinite variance. We introduce a smoothed empirical log-likelihood ratio statistic. Also, the paper proves a nonparametric version of Wilks’s theorem. Furthermore, we conduct a simulation to illustrate the performance of the proposed method.


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## Body

## 1. Introduction

Consider the stationary ARMA(p,q) time series {yt} generated by
(1)yt=φ1yt-1+⋯+φpyt-p+εt+ϑ1εt-1+⋯+ϑqεt-q,
where the innovation process {εt} is a sequence of i.i.d. random variables. When E(εt2)=∞, model (1) is an infinite variance autoregressive moving average (IVARMA) model, which defines a heavy-tailed process {yt}. For model (1), statistical inference has been explored in many studies (see, e.g., [1, 2]). Recently, for example, Pan et al. [3] and Zhu and Ling [4] proposed a weighted least absolute deviations estimator (WLADE) for model (1) and obtained the asymptotic normality.However, in the building of ARMA models, we are usually only interested in statistical inference for partial parameters. For example, in the sparse coefficient (a part of zero coefficients) ARMA models, it is necessary to determine which coefficient is zero. For model (1), one traditional method is to construct confidence regions for the partial parameters of interest by normal approximation as in [3]. However, since the limit distribution depends on the unknown nuisance parameters and density function of the errors, estimating the asymptotic variance is not a trivial task. Based on these, this paper tries to put forward a new method for the estimation of partial parameters of ARMA models. We propose an empirical likelihood method, which was introduced by Owen [5, 6]. Based on the estimating equations of WLADE, a smoothed profile empirical likelihood ratio statistic is derived, and a nonparametric version of Wilks’s theorem is proved. Therefore, we can construct confidence regions for the partial parameters of interest. Also, simulations suggest that, for relative small sample cases, the empirical likelihood confidence regions are more accurate than those confidence regions constructed by the normal approximation based on the WLADE proposed by Pan et al. [3].As an effective nonparametric inference method, the empirical likelihood method produces confidence regions whose shape and orientation are determined entirely by the data and therefore avoids secondary estimation. In the past two decades, the empirical likelihood method has been extended to many applications [7]. There are also many studies of empirical likelihood method for autoregressive models. Monti [8] considered the empirical likelihood in the frequency domain; Chuang and Chan [9] developed the empirical likelihood for unstable autoregressive models with innovations being a martingale difference sequence with finite variance; Chan et al. [10] applied the empirical likelihood to near unit root AR(1) model with infinite variance errors; Li et al. [11, 12], respectively, used the empirical likelihood to infinite variance AR(p) models and model (1).The rest of the paper is organized as follows. In Section2, we propose the profile empirical likelihood for the parameters of interest and show the main result. Section 3 provides the proofs of the main results. Some simulations are conducted in Section 4 to illustrate our approach. Conclusions are given in Section 5.


## 2. Methodology and Main Results

First, the parameter space is denoted byΘ⊂Rp+q, which contains the true value θ0 of the parameter θ as an inner point. For θ=(φ1,…,φp,ϑ1,…,ϑq), put
(2)εt(θ)={yt-∑i=1p‍φiyt-i-∑j=1q‍ϑjεt-j(θ),ift>0,0,ift≤0,
where yt≡0 for all t≤0, and note that εt(θ0)≠εt, because of this truncation.We define the objective function as(3)Sn(θ)=∑t=u+1n‍w~t|εt(θ)|,
where u≥max⁡(p,q) and the weight function w~t=1/(1+∑k=1t-1‍k-α|yt-k|)4, depending on a constant α>2. The WLADE, denoted by θ^, is a lacol minimizer of Sn(θ) in a neighborhood of θ0 [3]. Denote At(θ)=(At,1(θ),…,At,p+q(θ))τ, where At,i(θ)=-∂εt(θ)/∂θi. By (8.11.9) of Brockwell and Davis [13], it holds for t>max⁡(p,q) that
(4)At,i(θ)+∑j=1q‍ϑjAt-j,i(θ)=yt-i,i=1,…,p,At,i+p(θ)+∑j=1q‍ϑjAt-j,i+p(θ)=εt-i(θ),i=1,…,q.
Hence, θ^ satisfies estimating equation
(5)1n-u∑t=u+1n‍w~tAt(θ)sgn⁡{εt(θ)}=0,
where sgn⁡(x)=-1 for x<0 and =1 for x≥0 (see [14]). Note that the above estimating equation is not differentiable at point θ such that εt(θ)=0 for some t. This causes some problems for our subsequent asymptotic analysis. To overcome this problem, we replace it with a smooth function. Define a probability density kernel K(·) [15] such that ∫-∞+∞‍xjK(x)dx=0,κ for j=1,2, respectively, where κ≠0. Let Gh(x)=∫-x/hx/h‍K(u)du for h>0. Then, a smoothed version of (5) is
(6)1n-u∑t=u+1n‍w~tAt(θ)Gh(εt(θ))=0.Letmth(θ)=w~tAt(θ)Gh(εt(θ)); a smoothed empirical log-likelihood ratio is defined as
(7)lh(θ)=-2sup⁡{∑t=u+1n‍log⁡⁡((n-u)pt)∣∑t=u+1n‍ptmth(θ)=0,nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnpt≥0,∑t=u+1n‍pt=1}.
Using the Lagrange multiplier, the optimal value of pt is derived to be
(8)pt(θ)=1(n-u)(1+λ(θ)τmth(θ)),nnnnnnnnnnnnnu+1≤t≤n,
where λ(θ) is a p+q-dimensional vector of Lagrange multipliers satisfying
(9)1n-u∑t=u+1n‍mth(θ)1+λ(θ)τmth(θ)=0.
This gives the smoothed empirical log-likelihood ratio statistic
(10)lh(θ)=2∑t=u+1n‍log⁡(1+λ(θ)τmth(θ)).Letθ=(ϕτ,ωτ)τ, where ω∈Rm (1≤m≤p+q) is the parameter of interest and ϕ∈Rp+q-m is the nuisance parameter. Note that m=p+q means no nuisance parameters. Let ϕ0 and ω0 denote the true values of ϕ and ω, respectively. The profile empirical likelihood is defined as
(11)lp(ω)=min⁡ϕlh(ϕ,ω).
That is, lp(ω)=lh(ϕ~(ω),ω), where ϕ~=ϕ~(ω)≔argmin⁡ϕlh(ϕ,ω).The following conditions are in order.(A1)
The characteristic polynomialϕ(z)=1-φ1z-⋯-φpzp and θ(z)=1+ϑ1z+⋯+ϑqzq have no common zeros, and all roots of ϕ(z) and θ(z) are outside the unit circle.(A2)
The innovation{εt} has zero median and a differentiable density f(x) satisfying the conditions f(0)>0, sup⁡x∈R|f(x)|<B1<∞, and sup⁡x∈R|f′(x)|<B2<∞. Furthermore, E|εt|δ<∞ for some δ>0, and α>max⁡{2,2/δ}.(A3)
Asn→∞, u→∞ and u/n→0.(A4)
The second derivative ofK exists in R and K′(x) and K′′(x) are bounded.(A5)
h=1/nγ with 1/4<γ<1/3.First we show the existence and consistency ofϕ~(ω0).Proposition 1.
Letdn=1/nβ with max⁡{1/3,3γ/2}<β<1/2. Assume (A1)–(A5) hold; then as n→∞, with probability 1, there exists a local minimizer ϕ~ of lh(ϕ,ω0) which lies in the interior of the ball B={ϕ:∥ϕ-ϕ0∥≤dn}. Moreover ϕ~ and λ~=λ(ϕ~,ω0) satisfy
(12)Q1n(ϕ~,λ~)=0,Q2n(ϕ~,λ~)=0,
where
(13)Q1n(ϕ,λ)=1n-u∑t=u+1n‍mth(ϕ,ω0)1+λτmth(ϕ,ω0),Q2n(ϕ,λ)=1n-u∑t=u+1n‍11+λτmth(ϕ,ω0)(∂mth(ϕ,ω0)∂ϕτ)τλ.The following theorem presents the asymptotic distribution of the profile empirical likelihood.Theorem 2.
Under conditions of Proposition1, as n→∞, the random variable lp(ω0), with ϕ~ given in Proposition 1, converges in distribution to χm2.Ifc is chosen such that P(χm2≤c)=a, then Theorem 2 implies that the asymptotic coverage probability of empirical likelihood confidence region Ihc=(ω:lp(ω)≤c) will be a; that is, P(ω0∈Ihc)=P(lp(ω0)≤c)=a+o(1), as n→∞.


## 3. Proofs of the Main Results

In the following,∥·∥ denotes the Euclidian norm for a vector or matrix and C denotes a positive constant which may be different at different places. For t=0,±1,±2,…, define
(14)Ut-∑i=1p‍φi0Ut-i=εt,Vt+∑j=1q‍ϑj0Vt-j=εt.
Put Qt=(Ut-1,…,Ut-p,Vt-1,…,Vt-q)τ, wt=1/(1+∑k=1∞‍k-α|yt-k|)4, and the corresponding partial vector for ϕ0 is denoted by Q1t. Let
(15)Σ=E(wtQtQtτ),Σ1=E(wtQtQ1tτ),Ω=E(wt2QtQtτ).
Assumptions A1 and A2 imply that, for δ~=min⁡(δ,1),
(16)E(∑k=1∞‍k-α/2|yt-k|)δ~≤∑k=1∞‍k-αδ~/2E|yt-k|δ~<∞.
Hence, ∑k=1∞‍k-α/2|yt-k|<∞ with probability 1, which ensures that wt is well defined. Note that ∥Qt∥≤C∑j=1∞‍rj|yt-j| for some 0<r<1 and
(17)wt1/2∥Qt∥≤C∑j=1∞‍rj|yt-j|1+∑k=1∞‍k-α|yt-k|≤C∑k=1∞‍rkkα<∞.
Then, Σ, Σ1, and Ω are well-defined (finite) matrices. For simplicity, we denote (ϕ,ω0) and (ϕ0,ω0) by ϕ and ϕ0, respectively, in this section. The following notations will be used in the proofs. Let
(18)Zn(ϕ)=max⁡u+1≤t≤n∥mth(ϕ)∥,Qnh(ϕ)=1n-u∑t=u+1n‍mth(ϕ),S(ϕ)=1n-u∑t=u+1n‍mth(ϕ)mth(ϕ)τ.
To prove Proposition 1, we first prove the following lemmas.Lemma 3.
Under the conditions of Proposition1, as n→∞,
(19)(i)Qnh(ϕ0)=O(log⁡nn)a.s.,(ii)∂Qnh(ϕ0)∂ϕτ=-2f(0)Σ1+o(1)a.s.Proof of Lemma3.
For part (i), we may write(20)n-uQnh(ϕ0)=1n-u∑t=u+1n‍wtQtGh(εt)+1n-u∑t=u+1n‍[wtQt(Gh(εt(ϕ0))-Gh(εt))nnnnnnnnnnnnnnnn+wt(At(ϕ0)-Qt)Gh(εt(ϕ0))(Gh(εt(ϕ0))-Gh(εt))]+1n-u∑t=u+1n‍(w~t-wt)At(ϕ0)Gh(εt(ϕ0))=:K1+K2+K3.
For K1, we have
(21)K1n-u=1n-u∑t=u+1n‍Ztbnt+(1n-u∑t=u+1n‍Zt)O(h2),
where Zt=wtQt, bnt=Gh(εt)-E(Gh(εt)). The second term of (21) is O(log⁡n/n) a.s. by the ergodicity. Now turning to the first term, we suppose that Qt is the first element Ut-1 without loss of generality. Note that, for each n≥u+1, {Ztbnt,Ft,u+1≤t≤n} is a sequence of martingale differences with |Ztbnt|≤C, where Ft=σ(εs,s≤t). For some C0>0, by the ergodicity, we have
(22)Vn2=∑t=u+1n‍E{(Ztbnt)2∣Ft-1}=∑t=u+1n‍Zt2E(bnt)2≤C∑t=u+1n‍Zt2<C(E(Zt2)+C0)na.s.
Set y=C(E(Zt2)+C0)n; by Theorem 1.2A in [16], for all A>0, we have
(23)P{|∑t=u+1n‍Ztbnt|>Anlog⁡n}=P{|∑t=u+1n‍Ztbnt|>Anlog⁡n,Vn2<yforsomen}≤2exp⁡⁡{-A2nlog⁡n2(y+CAnlog⁡n)}=2exp⁡{-A2log⁡n2C(E(Zt2)+C0)+2CAlog⁡n/n}.
Choosing A such that A2>2C(E(Zt2)+C0), by the Borel-Cantelli lemma, the first term of (21) is O(log⁡n/n) a.s. Thus, K1 is O(log⁡n)a.s. For K2, by Davis [2], it holds that |εt-εt(ϕ0)|≤ξt, and ∥At(ϕ0)-Qt∥≤ξt, where ξt=C∑j=t∞‍rj|yt-j| for some 0<r<1. Therefore,
(24)∥K2∥≤Chn-u∑t=u+1n‍wt∥Qt∥|εt-εt(ϕ0)|+Cn-u∑t=u+1nwt∥Qt-At(ϕ0)∥≤Chn-u∑t=u+1nξt+Cn-u∑t=u+1nξt=Chn-u∑t=u+1nrt∑l=0∞rl|y-l|+Cn-u∑t=u+1nrt∑l=0∞rl|y-l|⟶a.s.0.
Thus, K2 is O(log⁡n)a.s. For K3, we have
(25)∥K3∥≤Cn-u∑t=u+1n‍|wt-w~t|∥At(ϕ0)∥≤Cn-u∑t=u+1n‍∑k=t∞‍k-α|yt-k|∑j=1t-1‍rj|yt-j|1+∑k=1t-1‍k-α|yt-k|≤Cn-u∑t=u+1n‍∑k=t∞‍k-α|yt-k|∑j=1t-1‍rjjα≤Cn-u∑t=u+1n‍∑k=t∞‍k-α|yt-k|≤Cn-u∑t=u+1n‍∑l=0∞‍(t+l)-α|y-l|≤Cn-u∑t=u+1n‍(t-α|y0|+∑l=1∞‍2-αt-α/2l-α/2|y-l|)=Cn-u∑t=u+1n‍t-α|y0|+Cn-u∑t=u+1n‍t-α/2∑l=1∞‍l-α/2|y-l|⟶a.s.0
because we have the facts that ∥At(ϕ0)∥≤C∑j=1t-1‍rj|yt-j| (see [2]) and (t+l)-α≤2-α(tl)-α/2 for t>0, l>0. Thus, K3 is also O(log⁡n)a.s. Therefore part (i) holds. For the proof of part (ii), we may write
(26)∂Qnh(ϕ0)∂ϕτ=-1(n-u)h∑t=u+1n‍w~tAt(ϕ0)A1tτ(ϕ0)×[K(εt(ϕ0)h)+K(-εt(ϕ0)h)]+1n-u∑t=u+1n‍w~t∂At(ϕ0)∂ϕτGh(εt(ϕ0))=D1+D2,
where A1t(ϕ)=-∂εt(ϕ)/∂ϕ. For D1, we may write
(27)D1=-1(n-u)h∑t=u+1n‍wtQtQ1tτ(K(εth)+K(-εth))-1(n-u)h∑t=u+1n‍[(K(εth)+K(-εth))w~tAt(ϕ0)A1tτ(ϕ0)nnnnnnnnnnnnnnnn×(K(εt(ϕ0)h)+K(-εt(ϕ0)h))nnnnnnnnnnnnnnnn-wtQtQ1tτ(K(εth)+K(-εth))]=D11+D12.
Note that
(28)D11=-1(n-u)h∑t=u+1n‍Ttcnt-1(n-u)h×∑t=u+1n‍wtQtQ1tτE(K(εth)+K(-εth))=-1(n-u)h∑t=u+1n‍Ttcnt-[1n-u∑t=u+1n‍wtQtQ1tτ](2f(0)+o(h)),
where Tt=wtQtQ1tτ and cnt=K(εt/h)+K(-εt/h)-E[K(εt/h)+K(-εt/h)]. The second term of (28) is -2f(0)Σ1 a.s. by the ergodicity. We will prove that the first term is o(1) a.s. We suppose that Qt is the first element Ut-1 without loss of generality. Note that, for each n≥u+1, {Ttcnt,Ft,u+1≤t≤n} is a sequence of martingale differences with |Ttcnt|≤C, and
(29)Vn2=∑t=u+1n‍E{(Ttcnt)2∣Ft-1}=∑t=u+1n‍Tt2E(cnt)2<nC(f(0)C0h+O(h2))a.s.,
where C0>0 is a constant. Set y=nC(f(0)C0h+O(h2)); by Theorem 1.2A in [16], for all ε>0, we have
(30)P{|∑t=u+1n‍Ttcnt|>(nh)ε}=P{|∑t=u+1n‍Ttcnt|>(nh)ε,Vn2<yforsomen}≤2exp⁡{-(nh)2ε22(y+Cnhε)}=2exp⁡{-nhε22C(f(0)C0+O(h))+2Cε}.
The result follows from the Borel-Cantelli lemma. Thus D11=-2f(0)Σ1+o(1) a.s. Similar to K2 and K3, we have
(31)∥D12∥≤1(n-u)h∑t=u+1n‍wt(∥Qt-At(ϕ0)∥∥Q1t∥nnnnnnnnnnnnnnnnnn+∥At(ϕ0)∥∥Q1t-A1t(ϕ0)∥)|K(εt(ϕ0)h)+K(-εt(ϕ0)h)|+1(n-u)h∑t=u+1n‍|wt-w~t|∥At(ϕ0)∥∥A1t(ϕ0)∥×|K(εt(ϕ0)h)+K(-εt(ϕ0)h)|+1(n-u)h∑t=u+1nwt∥Qt∥∥Q1t∥×|-εt(ϕ0)h[K(εth)+K(-εth)]nnnnnn-[K(εt(ϕ0)h)+K(-εt(ϕ0)h)]|≤C(n-u)h∑t=u+1n‍wt(∥Qt-At(ϕ0)∥∥Q1t∥nnnnnnnnnnnnnnnnnn+∥At(ϕ0)∥∥Q1t-A1t(ϕ0)∥)+C(n-u)h∑t=u+1n‍|wt-w~t|∥At(ϕ0)∥∥A1t(ϕ0)∥+C(n-u)h2∑t=u+1n‍wt∥Qt∥∥Q1t∥|εt-εt(ϕ0)|≤C(n-u)h∑t=u+1nξt+C(n-u)h∑t=u+1n∑k=t∞k-α|yt-k|+C(n-u)h2∑t=u+1nξt⟶a.s.0.
Therefore, D1=-2f(0)Σ1+o(1)a.s. For D2, from the definition of At(ϕ), it holds for t>max⁡(p,q) that
(32)θ(B)∂At,i(ϕ)∂ϕj=0,i,j=1,…,p,θ(B)∂At,i(ϕ)∂ϕj+p=-At-j,i(ϕ),i=1,…,p,j=1,…,q,θ(B)∂At,j+p(ϕ)∂ϕi=-At-j,i(ϕ),i=1,…,p,j=1,…,q,θ(B)∂At,i+p(ϕ)∂ϕj+p=-At-j,i+p(ϕ)-At-i,j+p(ϕ),nnnnnnnnnnnnnnnnnnnnnnnnnnnni,j=1,…,q,
where B is the backshift operator. For t=0,±1,±2,…, define
(33)θ0(B)Xt(i,j)=0,i,j=1,…,p,θ0(B)Xt(i,j+p)=-Qt-j,i,i=1,…,p,j=1,…,q,θ0(B)Xt(j+p,i)=-Qt-j,i,i=1,…,p,j=1,…,q,θ0(B)Xt(i+p,j+p)=-Qt-j,i+p-Qt-i,j+p,nnnnnnnnnnnnnnnnnnnni,j=1,…,q,
where Qt,i is the ith component of Qt. Put Xt=(Xt(i,j)); similar to [13], we have that ∥∂At(ϕ0)/∂ϕτ∥≤C∑j=1t-1‍rj|yt-j|, ∥Xt∥≤C∑j=1∞‍rj|yt-j|, and ∥Xt-∂At(ϕ0)/∂ϕτ∥≤ξt. Then, we may write
(34)D2=1n-u∑t=u+1n‍wtXtGh(εt)+1n-u∑t=u+1n‍[w~t∂At(ϕ0)∂ϕτGh(εt(ϕ0))-wtXtGh(εt)]=D21+D22.
Similar to D11 and D12, we have that D21 and D22 are o(1) a.s. This completes the proof.Lemma 4.
Under the conditions of Proposition1, as n→∞,
(35)(i)Zn(ϕ)=o(n1/3)a.s.,(ii)S(ϕ)=Ω+o(1)a.s.
hold uniformly in B.Proof of Lemma4.
For part (i), from [2], we have that ∥∂At(ϕ)/∂ϕτ∥≤C∑j=1t-1‍rj|yt-j| and ∥At(ϕ)∥≤C∑j=1t-1‍rj|yt-j| uniformly hold in the ball B for sufficiently large n. Then, for each ϕ∈B, we have
(36)∥mth(ϕ)∥≤w~t∥At(ϕ)∥≤C∑j=1t-1‍rj|yt-j|1+∑k=1t-1‍k-α|yt-k|≤C∑k=1t-1‍rkkα≤C∑k=1∞‍rkkα<∞.
Thus, part (i) holds. For part (ii), similar to the proof of Lemma 3, we have that S(ϕ0)=Ω+o(1)a.s. For each ϕ∈B, by Taylor expansion, we have
(37)S(ϕ)-S(ϕ0)=1n-u∑t=u+1n‍mth(ϕ0){∂mth(ϕ*)∂ϕτ(ϕ-ϕ0)}τ+1n-u∑t=u+1n‍{∂mth(ϕ*)∂ϕτ(ϕ-ϕ0)}mthτ(ϕ0)+1n-u∑t=u+1n‍{∂mth(ϕ*)∂ϕτ(ϕ-ϕ0)}×{∂mth(ϕ*)∂ϕτ(ϕ-ϕ0)}τ=:T1+T2+T3,
where ϕ* lies between ϕ0 and ϕ. For T1, we have
(38)∥T1∥≤1n-u∑t=u+1n‍w~t2∥At(ϕ0)∥∥∂At(ϕ*)∂ϕτ∥∥ϕ-ϕ0∥+1(n-u)h∑t=u+1n‍w~t2∥At(ϕ0)∥∥At(ϕ*)∥nnnnnnnnnnnnnnn×∥A1t(ϕ*)∥∥ϕ-ϕ0∥nnnnnnnnnnnnnnn×|K(εt(ϕ*)h)+K(-εt(ϕ*)h)|≤Cdnn-u∑t=u+1n‍(∑j=1t-1‍rjjα)2+Cdn(n-u)h∑t=u+1n‍(∑j=1t-1‍rjjα)3⟶a.s.0.
Similarly, we have T2→a.s.0 and T3→a.s.0. This completes the proof.Proof of Proposition1.
Forϕ∈B, by Taylor expansion,
(39)Qnh(ϕ)=Qnh(ϕ0)+∂Qnh(ϕ0)∂ϕτ(ϕ-ϕ0)+12∑j,k=1p+q-m‍∂2Qnh(ϕ0)∂ϕj∂ϕk(ϕj-ϕj0)(ϕk-ϕk0)+16∑j,k,l=1p+q-m‍∂3Qnh(ϕ*)∂ϕj∂ϕk∂ϕlnnnnnnnnn×(ϕj-ϕj0)(ϕk-ϕk0)(ϕl-ϕl0),
where ϕ* lies between ϕ0 and ϕ. Note that the final term on the right side of (39) can be written as
(40)16∑j,k,l=1p+q-m‍[(ϕj-ϕj0)(ϕk-ϕk0)(ϕl-ϕl0)h3]×(h3n-u∑t=u+1n‍∂3mth(ϕ*)∂ϕj∂ϕk∂ϕl),
which is o(δn) a.s., where δn=∥ϕ-ϕ0∥, because dn2/h3=1/n2β-3γ→0 and
(41)∥h3n-u∑t=u+1n‍∂3mth(ϕ*)∂ϕj∂ϕk∂ϕl∥≤C.
The third term on the right side of (39) can be written as
(42)12∑j,k=1p+q-m‍[(ϕj-ϕj0)(ϕk-ϕk0)h]nnnnn×{hn-u∑t=u+1n‍∂2mth(ϕ0)∂ϕj∂ϕk},
which is also o(δn) a.s., because dn/h=1/nβ-γ→0, and
(43)hn-u∑t=u+1n‍∂2mth(ϕ0)∂ϕj∂ϕk=o(1)a.s.,
by a similar proof of Lemma 3. Therefore,
(44)Qnh(ϕ)=Qnh(ϕ0)+∂Qnh(ϕ0)∂ϕτ(ϕ-ϕ0)+o(δn)a.s.,
uniformly about ϕ∈B. Denote ϕ=ϕ0+μdn, for ϕ∈{ϕ:∥ϕ-ϕ0∥=dn}, where ∥μ∥=1. Now, we give a lower bound for lh(ϕ) on the surface of the ball. Similar to [6], by Lemmas 3 and 4, we have
(45)lh(ϕ)=(n-u)Qnh(ϕ)τS(ϕ)-1Qnh(ϕ)+o(n4/3-3β)a.s.=(n-u)[O(log⁡nn)nnnnnnnnnnn+(-2f(0))Σ1μdn+o(dn)(log⁡nn)]τΩ-1×[O(log⁡nn)+(-2f(0))Σ1μdn+o(dn)]+o(n4/3-3β)a.s.≥(c-ε)n1-2βa.s.,
where c-ε>0 and c is the smallest eigenvalue of 4f2(0)Σ1τΩ-1Σ1. Similarly,
(46)lh(ϕ0)=(n-u)Qnh(ϕ0)τS(ϕ0)-1Qnh(ϕ0)+o(1)a.s.=O(log⁡n)a.s.
Since lh(ϕ) is a continuous function about ϕ as ϕ belongs to the ball B, lh(ϕ) attains its minimum value at some point ϕ~ in the interior of this ball, and ϕ~ satisfies ∂lh(ϕ~)/∂ϕ=0, it follows that (12) holds. This completes the proof.Proof of Theorem2.
Similar to the proof of Theorem2 of Qin and Lawless [17], we have
(47)(λ~ϕ~-ϕ0)=Sn-1(-Q1n(ϕ0,0)+op(n-1/2)op(n-1/2)),
where
(48)Sn=(∂Q1n(ϕ0,0)∂λτ∂Q1n(ϕ0,0)∂ϕτ∂Q2n(ϕ0,0)∂λτ0)⟶p(S11S12S210)=(-Ω-2f(0)Σ1-2f(0)Σ1τ0).
By the standard arguments in the proof of empirical likelihood (see [6]), we have
(49)lp(ω0)=-(n-u)(Qnh(ϕ0)τ,0)Sn-1(Qnh(ϕ0)τ,0)τ+op(1)=(Ω-1/2n-uQnh(ϕ0))τ×(I-4f2(0)Ω-1/2Σ1Δ-1Σ1τΩ-1/2)×(Ω-1/2n-uQnh(ϕ0))+op(1),
where Δ=4f2(0)Σ1τΩ-1Σ1. Since n-uQnh(ϕ0)→dN(0,Ω) and
(50)tr⁡{4f2(0)Ω-1/2Σ1Δ-1Σ1τΩ-1/2}=tr⁡{Δ-14f2(0)Σ1τΩ-1Σ1}=p+q-m,
it follows that lp(ω0)→dχm2.


## 4. Simulation Studies

We generated data from a simple ARMA(1, 1) modelyt=φ1yt-1+εt+ϑ1εt-1, with N(0,1), t2, and Cauchy innovation distribution. We set u=20, α=3, and the true value (φ1,ϑ1)=(0.4,0.7) or (-0.5,0.7), where φ1 is the parameter of interest. The sample size n=50,100,150,200, and 2,000 replications are conducted in all cases. We smooth the estimating equations using kernel
(51)K(x)=12πσe-x2/2σ2,
where σ=0.1, which is the so-called Gaussian kernel. The coverage probabilities of smoothed empirical likelihood confidence regions Ihc with the bandwidth h=1/nγ are denoted by EL(γ), where γ=0.27,0.30,0.32, respectively.As another benchmark of the simulation experiments, we consider the confidence regions based on the asymptotic normal distribution of WLADE proposed by [3]. To construct the confidence regions, we need to estimate f(0), Σ, and Ω. We can estimate f(0) by
(52)f^(0)=1σ^wbn(n-u)∑t=u+1n‍w~tK~{εt(θ^)bn},
where K~(x)=exp⁡(-x)/(1+exp⁡(-x))2 is a kernel function on R and bn=1/nν is a bandwidth, σ^w=(n-u)-1∑t=u+1n‍w~t. Σ and Ω can be estimated, respectively, by
(53)Σ^=1n-u∑t=u+1n‍w~tQ^tQ^tτ,Ω^=1n-u∑t=u+1n‍w~t2Q^tQ^tτ,
where Q^t is defined in the same manner as Qt, θ0 is replaced by θ^, and εt is replaced by εt(θ^); see (14). Based on this, we can construct a NA confidence region (i.e., based on the normal approximation of WLADE). The coverage probabilities of confidence regions INA based on the bandwidth bn=1/nν are denoted by NA(ν), with ν=0.25,0.20, respectively. Tables 1, 2, and 3 show the probabilities of the confidence intervals of φ1 at confidence levels 0.9 and 0.95, respectively.Table 1
The coverage probability of confidence intervals whenεt~N(0,1).
n
EL(0.27)
EL(0.30)
EL(0.32)
NA(0.25)
NA(0.20)
a
=
0.9   φ1=0.4
50
0.8818
0.8820
0.8822
0.8193
0.7875
100
0.8898
0.8896
0.8897
0.8655
0.8431
150
0.8926
0.8927
0.8932
0.8692
0.8395
200
0.8983
0.8983
0.8986
0.8666
0.8363
a
=
0.9  φ1=-0.5
50
0.8888
0.8885
0.8892
0.8156
0.7813
100
0.8967
0.8968
0.8973
0.8738
0.8448
150
0.8943
0.8944
0.8946
0.8823
0.8574
200
0.8972
0.8979
0.8977
0.8931
0.8692
a
=
0.95 φ1=0.4
50
0.9347
0.9350
0.9350
0.8724
0.8425
100
0.9424
0.9430
0.9431
0.9123
0.8862
150
0.9467
0.9471
0.9470
0.9157
0.8936
200
0.9494
0.9494
0.9497
0.9160
0.8937
a
=
0.95 φ1=-0.5
50
0.9404
0.9404
0.9404
0.8705
0.8407
100
0.9472
0.9474
0.9474
0.9134
0.8931
150
0.9481
0.9479
0.9476
0.9248
0.9052
200
0.9495
0.9495
0.9490
0.9326
0.9152Table 2
The coverage probability of confidence intervals whenεt~t2.
n
EL(0.27)
EL(0.30)
EL(0.32)
NA(0.25)
NA(0.20)
a
=
0.9  φ1=0.4
50
0.8774
0.8776
0.8779
0.7627
0.7290
100
0.8850
0.8849
0.8856
0.8323
0.7995
150
0.8919
0.8917
0.8924
0.8467
0.8157
200
0.8935
0.8932
0.8932
0.8510
0.8229
a
=
0.9  φ1=-0.5
50
0.8902
0.8902
0.8899
0.7295
0.7055
100
0.8942
0.8944
0.8950
0.8151
0.7836
150
0.8946
0.8941
0.8937
0.8473
0.8204
200
0.8965
0.8963
0.8961
0.8641
0.8392
a
=
0.95 φ1=0.4
50
0.9331
0.9327
0.9327
0.8163
0.7854
100
0.9412
0.9412
0.9413
0.8834
0.8542
150
0.9447
0.9446
0.9447
0.8965
0.8718
200
0.9456
0.9458
0.9459
0.9001
0.8762
a
=
0.95 φ1=-0.5
50
0.9421
0.9418
0.9418
0.7963
0.7658
100
0.9455
0.9462
0.9460
0.8695
0.8430
150
0.9444
0.9442
0.9440
0.8910
0.8695
200
0.9464
0.9463
0.9464
0.9072
0.8863Table 3
The coverage probability of confidence intervals whenεt~Cauchy.
n
EL(0.27)
EL(0.30)
EL(0.32)
NA(0.25)
NA(0.20)
a
=
0.9  φ1=0.4
50
0.8360
0.8358
0.8361
0.6345
0.6018
100
0.8708
0.8708
0.8702
0.7286
0.6942
150
0.8811
0.8819
0.8823
0.7614
0.7270
200
0.8870
0.8864
0.8868
0.7800
0.7467
a
=
0.9  φ1=-0.5
50
0.8673
0.8679
0.8678
0.5818
0.5502
100
0.8869
0.8870
0.8874
0.6756
0.6440
150
0.8972
0.8974
0.8974
0.7232
0.6867
200
0.8956
0.8955
0.8953
0.7510
0.7156
a
=
0.95 φ1=0.4
50
0.8992
0.8994
0.8996
0.6888
0.6557
100
0.9263
0.9261
0.9263
0.7870
0.7530
150
0.9362
0.9366
0.9367
0.8176
0.7857
200
0.9402
0.9398
0.9399
0.8350
0.8044
a
=
0.95 φ1=-0.5
50
0.9235
0.9239
0.9240
0.6378
0.6065
100
0.9422
0.9420
0.9420
0.7342
0.7030
150
0.9474
0.9479
0.9481
0.7820
0.7530
200
0.9467
0.9471
0.9470
0.8057
0.7778The simulation results can be summarized as follows. The coverage probabilities of NA(ν) are much smaller than the nominal levels and very sensitive to the choice of bandwidth bn and εt. On the other hand, the coverage probabilities of EL(γ) are much better and less sensitive to the choice of bandwidth h and εt. As the sample size n increases, the coverage probabilities for both increase to the nominal levels, as one might expect.


## 5. Conclusions

This paper explores a profile empirical likelihood method to construct confidence regions for the partial parameters of interest in IVARMA models. We started with the foundation of estimating equations of WLADE; then from there, we derived smoothed empirical likelihood. Moreover, we have proved that the resulting statistics has asymptotic standard chi-squared distribution. Hence there is no need to estimate any additional quantity such as the asymptotic variance. The simulations indeed show that the proposed method has a good finite sample behavior, which experimentally confirms our method.



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*Source: 868970-2014-09-02.xml*