Subsampling and Size Limit

Subsampling and Size Limit - THE LIMIT OF FINITE-SAMPLE...

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THE LIMIT OF FINITE-SAMPLE SIZE AND A PROBLEM WITH SUBSAMPLING By Donald W. K. Andrews and Patrik Guggenberger March 2007 Revised July 2007 COWLES FOUNDATION DISCUSSION PAPER NO. 1605R COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/
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TheL im ito fF in ite-Samp leS ize and a Problem with Subsampling Donald W. K. Andrews 1 Cowles Foundation for Research in Economics Yale University Patrik Guggenberger Department of Economics UCLA June 2005 Revised: July 2007 1 Andrews gratefully acknowledges the research support of the National Science Founda- tion via grant number SES-0417911. Guggenberger gratefully acknowledges research support from a faculty research grant from UCLA in 2005. For helpful comments, we thank Victor Chernozukhov, Russell Davidson, Hannes Leeb, Azeem Shaikh, and the participants at various seminars and conferences at which the paper was presented.
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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, b n <n bootstrap, and standard f xed critical value tests based on such a test statistic often have asymptotic size–de f ned as the limit of the f nite- 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 veri f ed in several examples. Analogous results are established for con f dence intervals. The results apply to tests and con f dence intervals (i) when a parameter may be near a boundary, (ii) for parameters de f ned by moment inequalities, (iii) based on super-e cient 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 identi f cation at some point(s) in the parameter space, such as models with weak instruments and threshold autoregressive models, (vii) in pre- dictive regression models with nearly-integrated regressors, (viii) for non-di f erentiable functions of parameters, and (ix) for di f erentiable functions of parameters that have zero f rst-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 uniden- ti f ed parameters that are bounded by moment inequalities, i.e., example (ii), certain subsample con f dence 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.
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Subsampling and Size Limit - THE LIMIT OF FINITE-SAMPLE...

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