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Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data-dependent convex combination of two identification-robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi-conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug-in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification-robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.