Abstract
This paper mainly discusses two issues about asymptotic expansions for the distributions of $$\chi ^2$$ -type test statistics. First, it is shown that the generalized empirical likelihood ratio, Wald-type, and score-type test statistics for a subvector hypothesis in the possibly over-identified moment restrictions are, in general, not Bartlett-correctable, except for the empirical likelihood ratio test statistic. Second, starting with the classical likelihood or the modern generalized empirical likelihood, the Bartlett-type corrected test statistics, with the bootstrap procedure, are proposed to achieve a higher-order accurate testing inference for the nonparametric setup as well as the parametric setup.
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