Generic results for establishing the asymptotic size of confidence sets and tests

• Published in: Journal of econometrics Vol. 218; no. 2; pp. 496 - 531
• Main Authors: Andrews, Donald W.K., Cheng, Xu, Guggenberger, Patrik
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2020; Elsevier Science Publishers; Elsevier Sequoia S.A
• Subjects: Asymptotic methods; Asymptotic size; Autoregressive model; Bootstrap method; Confidence set; Estimating techniques; Grid bootstrap; Identification; Inference; Innovations; Nonlinear systems; Regression analysis; Test; Uniform; Weak identification; Weak instruments; Weak instruments; Asymptotic size; Confidence set; Weak identification; C10; C12; Test; Grid bootstrap; Inference; Autoregressive model; Identification; Uniform
• DOI: 10.1016/j.jeconom.2020.04.027
• ISSN: 0304-4076

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 is a discontinuous function of a parameter. The results are illustrated in several 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 kth 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. In addition, as a simple running example, we consider a two-sided equal-tailed CI for the AR coefficient in an AR(1) model, which is a simplified version of the CI in Andrews and Guggenberger (2014).