COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1813

Generic Results for Establishing the Asymptotic Size of Confidence Sets and Tests

Donald W.K. Andrews, Xu Cheng and Patrik Guggenberger

August 2011

This paper provides a set of results that can be used to establish the asymptotic size and/or similarity in a uniform sense of confidence sets and tests. The results are generic in that they can be applied to a broad range of problems. They are most useful in scenarios where the pointwise asymptotic distribution of a test statistic has a discontinuity in its limit distribution.

The results are illustrated in three examples. These are: (i) the conditional likelihood ratio test of Moreira (2003) for linear instrumental variables models with instruments that may be weak, extended to the case of heteroskedastic errors; (ii) the grid bootstrap confidence interval of Hansen (1999) for the sum of the AR coefficients in a k-th order autoregressive model with unknown innovation distribution, and (iii) the standard quasi-likelihood ratio test in a nonlinear regression model where identification is lost when the coefficient on the nonlinear regressor is zero.

Keywords: Asymptotically similar, Asymptotic size, Autoregressive model, Confidence interval, Nonlinear regression, Test, Weak instruments

JEL Classification: C12, C18, C22, C26