Empirical Likelihood for Partial Parameters in ARMA Models with Infinite Variance

This paper proposes a profile empirical likelihood for the partial parameters in ARMA 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.

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 2 Journal of Applied Mathematics 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() models and model (1).
The rest of the paper is organized as follows.In Section 2, 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.
The following conditions are in order.
(A1) The characteristic polynomial () = 1 − where The following theorem presents the asymptotic distribution of the profile empirical likelihood.

Theorem 2. Under conditions of Proposition 1, as
If  is chosen such that ( 2  ≤ ) = , then Theorem 2 implies that the asymptotic coverage probability of empirical likelihood confidence region  ℎ = ( :   () ≤ ) will be ;

Proofs of the Main Results
In the following, ‖ ⋅ ‖ denotes the Euclidian norm for a vector or matrix and  denotes a positive constant which may be different at different places.For  = 0, ±1, ±2, . .., define Put 4 , and the corresponding partial vector for  0 is denoted by  1 .Let Assumptions A1 and A2 imply that, for δ = min(, 1), ( Hence, Then, Σ, Σ 1 , and Ω are well-defined (finite) matrices.For simplicity, we denote (,  0 ) and ( 0 ,  0 ) by  and  0 , respectively, in this section.The following notations will be used in the proofs.Let To prove Proposition 1, we first prove the following lemmas.
Proof of Proposition 1.For  ∈ , by Taylor expansion, where  * lies between  0 and .Note that the final term on the right side of (39) can be written as which is (  ) a.s., where The third term on the right side of (39) can be written as which is also (  ) a.s., because   /ℎ = 1/ − → 0, and by a similar proof of Lemma 3. Therefore, +  ( 4/3−3 ) a.s.
where  −  > 0 and  is the smallest eigenvalue of Since  ℎ () is a continuous function about  as  belongs to the ball ,  ℎ () attains its minimum value at some point φ in the interior of this ball, and φ satisfies  ℎ ( φ)/ = 0, it follows that (12) holds.This completes the proof.
Proof of Theorem 2. Similar to the proof of Theorem 2 of Qin and Lawless [17], we have where ) . ( By the standard arguments in the proof of empirical likelihood (see [6]), we have where it follows that   ( 0 )

Simulation Studies
We generated data from a simple ARMA(1, 1) model   =  1  −1 +   +  1  −1 , with (0, 1),  2 , and Cauchy innovation distribution.We set  = 20,  = 3, and the true value ( 1 ,  1 ) = (0.4,0.7) or (−0.5, 0.7), where  1 is the parameter of interest.The sample size  = 50, 100, 150, 200, and 2,000 replications are conducted in all cases.We smooth the estimating equations using kernel where  = 0.1, which is the so-called Gaussian kernel.The coverage probabilities of smoothed empirical likelihood confidence regions  ℎ with the bandwidth ℎ = 1/  are denoted by EL(), where  = 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 (0), Σ, and Ω.We can estimate (0) by where K() = exp(−)/(1 + exp(−)) 2 is a kernel function on  and   = 1/ ] is a bandwidth, σ = ( − ) −1 ∑  =+1 w .Σ and Ω can be estimated, respectively, by where Q is defined in the same manner as   ,  0 is replaced by θ, and   is replaced by   ( 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  NA based on the bandwidth   = 1/ ] are denoted by NA(]), with ] = 0.25, 0.20, respectively.Tables 1,2, and 3 show the probabilities of the confidence intervals of  1 at confidence levels 0.9 and 0.95, respectively.The simulation results can be summarized as follows.The coverage probabilities of NA(]) are much smaller than the nominal levels and very sensitive to the choice of bandwidth   and   .On the other hand, the coverage probabilities of EL() are much better and less sensitive to the choice of bandwidth ℎ and   .As the sample size  increases, the coverage probabilities for both increase to the nominal levels, as one might expect.

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.

Table 2 :
The coverage probability of confidence intervals when   ∼  2 .

Table 3 :
The coverage probability of confidence intervals when   ∼ ℎ.