Some Properties of Tests for Parameters That Can Be Arbitrarily Close to Being Unidentified

Confidence intervals for parameters that can be arbitrarily close to being unidentified are unbounded with positive probability (e.g. Dufour, Econometrica 65, pp. 1365-1387 and Pfanzagl, Journal of Statistical Planning and Inference, 75, pp. 9-20), and the asymptotic risks of their estimators are unbounded (Pötscher, Econometrica 70, pp. 1035-1065). We extend these “impossibility results” and show that all tests of size α concerning parameters that can be arbitrarily close to being unidentified have power that can be as small as α for any sample size even if the null and the alternative hypotheses are not adjacent. The results are proved for a very general framework that contains commonly used models. 1 Address for correspondence: Giovanni Forchini, Department of Econometrics and Business Statistics, Monash University, Clayton, Victoria 3800, Australia. E-mail: [email protected] 2 I thank Grant Hillier, Patrick Marsh, Don Poskitt and Richard Smith for useful and encouraging comments and discussions. I thank the editor John Stufken and a referee for very useful feedback. This research was partially supported by Australian Research Council grant DP0771445.