The Limit of Finite Sample Size and a Problem with SubsamplingPatrik Guggenberger, UCLA Abstract 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) for parameters that 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 autoregressive models with roots that may be close to unity, (vi) in predictive regression models with nearly-integrated regressors, (vii) in models with lack of identification at some point(s) in the parameter space, such as models with weak instruments and threshold autoregressive models, (viii) for non-differentiable functions of parameters, and (ix) for differentiable functions of parameters that have zero first-order derivative. Keywords: Asymptotic size, b < n bootstrap, finite sample size, over-rejection, size correction, subsample confidence interval, subsample test. |