The Limit of Finite-Sample Size and a Problem with Subsampling
This paper considers inference based on a test statistic that has a limit distribution that is discontinuous in a nuisance parameter or the parameter of interest. The paper shows that subsample, b n < n bootstrap, and standard ﬁxed critical value tests based on such a test statistic often have asymptotic size — deﬁned as the limit of the ﬁnite-sample size — that is greater than the nominal level of the tests. We determine precisely the asymptotic size of such tests under a general set of high-level conditions that are relatively easy to verify. The high-level conditions are veriﬁed in several examples. Analogous results are established for conﬁdence intervals. The results apply to tests and conﬁdence intervals (i) when a parameter may be near a boundary, (ii) for parameters deﬁned by moment inequalities, (iii) based on super-eﬀicient or shrinkage estimators, (iv) based on post-model selection estimators, (v) in scalar and vector autoregressive models with roots that may be close to unity, (vi) in models with lack of identiﬁcation at some point(s) in the parameter space, such as models with weak instruments and threshold autoregressive models, (vii) in predictive regression models with nearly-integrated regressors, (viii) for non-diﬀerentiable functions of parameters, and (ix) for diﬀerentiable functions of parameters that have zero ﬁrst-order derivative. Examples (i)-(iii) are treated in this paper. Examples (i) and (iv)-(vi) are treated in sequels to this paper, Andrews and Guggenberger (2005a, b). In models with unidentiﬁed parameters that are bounded by moment inequalities, i.e., example (ii), certain subsample conﬁdence regions are shown to have asymptotic size equal to their nominal level. In all other examples listed above, some types of subsample procedures do not have asymptotic size equal to their nominal level.
Andrews, Donald W.K. and Guggenberger, Patrik, "The Limit of Finite-Sample Size and a Problem with Subsampling" (2007). Cowles Foundation Discussion Papers. 1898.