Research Article | Open Access
Jinyu Li, Wei Liang, Shuyuan He, "Empirical Likelihood for Partial Parameters in ARMA Models with Infinite Variance", Journal of Applied Mathematics, vol. 2014, Article ID 868970, 10 pages, 2014. https://doi.org/10.1155/2014/868970
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.
Consider the stationary ARMA time series generated by where the innovation process is a sequence of i.i.d. random variables. When , model (1) is an infinite variance autoregressive moving average (IVARMA) model, which defines a heavy-tailed process . For model (1), statistical inference has been explored in many studies (see, e.g., [1, 2]). Recently, for example, Pan et al.  and Zhu and Ling  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 . 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. .
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 . There are also many studies of empirical likelihood method for autoregressive models. Monti  considered the empirical likelihood in the frequency domain; Chuang and Chan  developed the empirical likelihood for unstable autoregressive models with innovations being a martingale difference sequence with finite variance; Chan et al.  applied the empirical likelihood to near unit root AR 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.
2. Methodology and Main Results
First, the parameter space is denoted by , which contains the true value of the parameter as an inner point. For , put where for all , and note that , because of this truncation.
We define the objective function as where and the weight function , depending on a constant . The WLADE, denoted by , is a lacol minimizer of in a neighborhood of . Denote , where . By (8.11.9) of Brockwell and Davis , it holds for that Hence, satisfies estimating equation where for and =1 for (see ). Note that the above estimating equation is not differentiable at point such that for some . 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  such that for , respectively, where . Let for . Then, a smoothed version of (5) is
Let ; a smoothed empirical log-likelihood ratio is defined as Using the Lagrange multiplier, the optimal value of is derived to be where is a -dimensional vector of Lagrange multipliers satisfying This gives the smoothed empirical log-likelihood ratio statistic
Let , where () is the parameter of interest and is the nuisance parameter. Note that means no nuisance parameters. Let and denote the true values of and , respectively. The profile empirical likelihood is defined as That is, , where .
The following conditions are in order.(A1)The characteristic polynomial and have no common zeros, and all roots of and are outside the unit circle.(A2)The innovation has zero median and a differentiable density satisfying the conditions , , and . Furthermore, for some , and .(A3)As , and .(A4)The second derivative of exists in and and are bounded.(A5) with .
First we show the existence and consistency of .
Proposition 1. Let with . Assume (A1)–(A5) hold; then as , with probability 1, there exists a local minimizer of which lies in the interior of the ball . Moreover and satisfy where
The following theorem presents the asymptotic distribution of the profile empirical likelihood.
If is chosen such that , then Theorem 2 implies that the asymptotic coverage probability of empirical likelihood confidence region will be ; that is, , as .
3. 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 , define Put , , and the corresponding partial vector for is denoted by . Let Assumptions A1 and A2 imply that, for , Hence, with probability , which ensures that is well defined. Note that for some and Then, , , and are well-defined (finite) matrices. For simplicity, we denote and by and , respectively, in this section. The following notations will be used in the proofs. Let To prove Proposition 1, we first prove the following lemmas.
Lemma 3. Under the conditions of Proposition 1, as ,
Proof of Lemma 3. For part (i), we may write For , we have where , . The second term of (21) is a.s. by the ergodicity. Now turning to the first term, we suppose that is the first element without loss of generality. Note that, for each , is a sequence of martingale differences with , where . For some , by the ergodicity, we have Set ; by Theorem in , for all , we have Choosing such that , by the Borel-Cantelli lemma, the first term of (21) is a.s. Thus, is . For , by Davis , it holds that , and , where for some . Therefore, Thus, is . For , we have because we have the facts that (see ) and for , . Thus, is also . Therefore part (i) holds. For the proof of part (ii), we may write where . For , we may write Note that where and . The second term of (28) is a.s. by the ergodicity. We will prove that the first term is a.s. We suppose that is the first element without loss of generality. Note that, for each , is a sequence of martingale differences with , and where is a constant. Set ; by Theorem in , for all , we have The result follows from the Borel-Cantelli lemma. Thus a.s. Similar to and , we have Therefore, . For , from the definition of , it holds for that where is the backshift operator. For , define where is the th component of . Put ; similar to , we have that , , and . Then, we may write Similar to and , we have that and are a.s. This completes the proof.
Lemma 4. Under the conditions of Proposition 1, as , hold uniformly in .
Proof of Lemma 4. For part (i), from , we have that and uniformly hold in the ball for sufficiently large . Then, for each , we have Thus, part (i) holds. For part (ii), similar to the proof of Lemma 3, we have that . For each , by Taylor expansion, we have where lies between and . For , we have Similarly, we have and . This completes the proof.
Proof of Proposition 1. For , by Taylor expansion, where lies between and . Note that the final term on the right side of (39) can be written as which is a.s., where , because and The third term on the right side of (39) can be written as which is also a.s., because , and by a similar proof of Lemma 3. Therefore, uniformly about . Denote , for , where . Now, we give a lower bound for on the surface of the ball. Similar to , by Lemmas 3 and 4, we have where and is the smallest eigenvalue of . Similarly, Since is a continuous function about as belongs to the ball , attains its minimum value at some point in the interior of this ball, and satisfies , it follows that (12) holds. This completes the proof.
Proof of Theorem 2. Similar to the proof of Theorem 2 of Qin and Lawless , we have where By the standard arguments in the proof of empirical likelihood (see ), we have where . Since and it follows that .
4. Simulation Studies
We generated data from a simple ARMA(1, 1) model , with , , and Cauchy innovation distribution. We set , , and the true value or , where is the parameter of interest. The sample size , and 2,000 replications are conducted in all cases. We smooth the estimating equations using kernel where , which is the so-called Gaussian kernel. The coverage probabilities of smoothed empirical likelihood confidence regions with the bandwidth are denoted by EL(), where , respectively.
As another benchmark of the simulation experiments, we consider the confidence regions based on the asymptotic normal distribution of WLADE proposed by . To construct the confidence regions, we need to estimate , , and . We can estimate by where is a kernel function on and is a bandwidth, . and can be estimated, respectively, by where is defined in the same manner as , is replaced by , and is replaced by ; 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 based on the bandwidth are denoted by NA(), with , respectively. Tables 1, 2, and 3 show the probabilities of the confidence intervals of at confidence levels 0.9 and 0.95, respectively.