  The upper bounds on the coverage probabilities of the confidence regions
based on blockwise empirical likelihood [Kitamura (1997)] and nonstandard
expansive empirical likelihood [Nordman et al. (2013)] methods for time series
data are investigated via studying the probability for the violation of 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
the fixed-b asymptotics, which has been recently shown to provide a more
accurate approximation to the finite sample distribution than the conventional
chi-square approximation. Our theoretical and numerical findings suggest that
both the finite sample and large sample upper bounds for coverage probabilities
are strictly less than one and the blockwise empirical likelihood confidence
region can exhibit serious undercoverage when (i) the dimension of moment
conditions is moderate or large; (ii) the time series dependence is positively
strong; or (iii) the block size is large relative to sample size. A similar
finite sample coverage problem occurs for the nonstandard 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 illustration demonstrate the effectiveness of
our proposed remedies in terms of delivering confidence sets with more accurate
coverage.
