On the Coverage Bound Problem of Empirical Likelihood Methods for Time Series

Abstract

The upper bounds on the coverage probabilities of the confidence regions based on blockwise empirical likelihood and non-standard expansive empirical likelihood methods for time series data are investigated via studying the probability of violating the convex hull constraint. The large sample bounds are derived on the basis of the pivotal limit of the blockwise empirical log-likelihood ratio obtained under fixed b asymptotics, which has recently been shown to provide a more accurate approximation to the finite sample distribution than the conventional χ 2 -approximation. Our theoretical and numerical findings suggest that both the finite sample and the large sample upper bounds for coverage probabilities are strictly less than 1 and the blockwise empirical likelihood confidence region can exhibit serious undercoverage when the dimension of moment conditions is moderate or large, the time series dependence is positively strong or the block size is large relative to the sample size. A similar finite sample coverage problem occurs for non-standard expansive empirical likelihood.To alleviate the coverage bound problem, we propose to penalize both empirical likelihood methods by relaxing the convex hull constraint. Numerical simulations and data illustrations demonstrate the effectiveness of our proposed remedies in terms of delivering confidence sets with more accurate coverage. Some technical details and additional simulation results are included in on-line supplemental material.

Empirical Likelihood for a Long Range Dependent Process Subordinated to a Gaussian Process

“…Similarly to resampling techniques that use data blocks to capture time series dependence (cf. Politis et al , ; Lahiri, ), Kitamura () proposed the blockwise empirical likelihood (BEL) methodology for stationary SRD time series, which has proven to provide valid inference in several scenarios with SRD processes, including regression (Bravo, ; Qin and Li, ), quantile estimation (Cheng and Wong, ; Lei and Qin, ), count data (Wu and Cao, ), Markov chains (Harari‐Kermadec, ) and associated processes (Xiong and Lin ; Zhang ) along with generalized forms of BEL (Bravo ; Lin and Zhang, ; Otsu, ; Zhang and Shao, ) and extensions to improve coverage accuracy (cf. tapering, Nordman, ; fixed b ‐asymptotics, Zhang and Shao, ).…”

High Order Adjusted Block-wise Empirical Likelihood For Weakly Dependent Data

“…These three methods then have been extended and refined by subsequent research. For instance, Zhang and Shao (2016) extended the penalized empirical likelihood to weakly dependent data by using a fixed-b block-wise method. Emerson and Owen (2009) proposed a modified adjusted empirical likelihood method.…”

“…It is worth pointing out that most of the existing work have focused on independent data, and the aforementioned Zhang and Shao (2016) was the first paper to address the convex hull constraint for weakly dependent data with penalized empirical likelihood under the block-wise framework, which was introduced to empirical likelihood by Kitamura (1997). Recently Piyadi Gamage et al (2017) studied the adjusted empirical likelihood for time series models under the frequency domain empirical likelihood framework, which was introduced by Nordman and Lahiri (2006).…”