Some Properties of Tests for Possibly Unidentified Parameters
G. Forchini
It is well known that confidence intervals for weakly identified
parameters are unbounded with positive probability (e.g. Dufour, Econometrica
65, pp. 1365-1387 and Staiger and Stock, Econometrica 65, pp. 557-586), and
that the asymptotic risk of their estimators is unbounded (Pötscher, Econometrica
70, pp.1035-1065). In this note we extend these "impossibility results" and
show that uniformly consistent tests for weakly identified parameters do not
exist. We also show that all similar tests of size α < 1/2 concerning possibly
unidentified parameters have type II error probability that can be as large
as 1 - α.
Keywords: Similar tests, Consistent tests, Weak instruments, Identification
