Abstract
This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.
Highlights
The empirical likelihood (EL) method is originally designed to construct a confidence region only for independent data [1,2]
To the best of our knowledge, there have been some works applying adjusted EL (AEL) on the improvement of the precision of the EL-based confidence region, such as [15] and [16]. Under their proposed conventional adjustment level a max (1, log(n)/2), the precision of chi-square approximation distribution of AEL is not obviously enhanced. We propose another adjustment level to construct new AEL for the parameters in stationary short- and longmemory time series models with Gaussian noise
We propose the MEL method for estimating the parameters of stationary time series models
Summary
The empirical likelihood (EL) method is originally designed to construct a confidence region only for independent data [1,2]. (MEL) was proposed to improve the precision of the EL-based confidence regions by constructing a new pseudo data point [14]. To the best of our knowledge, there have been some works applying AEL on the improvement of the precision of the EL-based confidence region, such as [15] and [16] Under their proposed conventional adjustment level a max (1, log(n)/2), the precision of chi-square approximation distribution of AEL is not obviously enhanced. The remainder of this article is organized as follows; in Section 2, we present the proposed AEL and MEL ratio statistics for parameters in stationary ARMA and ARFIMA processes and deduce their asymptotical chi-square properties.
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