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, bn < 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.

Keywords: Asymptotic size, b < n bootstrap, Finite-sample size, Over-rejection, Size correction, Subsample confidence interval, Subsample test

JEL Classification: C12, C15