On the coverage bound problem of empirical likelihood methods for time series

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 fixedb asymptotics, which has recently been shown to provide a more accurate approximation to the finite sample distribution than the conventional χ2-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.