COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS
AT YALE UNIVERSITY
Box 208281
New Haven, CT 06520-8281
COWLES FOUNDATION DISCUSSION PAPER NO. 1605R
The Limit of Finite-Sample Size and a Problem with Subsampling
Donald W.K. Andrews and Patrik Guggenberger
March 2007
Revised July 2007
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 Numbers: C12, C15 |