On the Power Curves of the Conditional Likelihood Ratio and Related Tests for Instrumental Variables Regression with Weak Instruments

We show that the Likelihood Ratio (LR) statistic for testing the value of the coef- ficient β in a linear instrumental variables model with a single endogenous variable is identical to the t0(βL)2 statistic as proposed by Mills, Moreira, and Vilela (2014), where βL is the LIML estimator. This implies the equivalence of their conditional versions that are robust to weak instruments. From this result, properties of the power of the Conditional LR (CLR) test can be understood; in particular the asym- metric nature of the power curve as a function of the true value of β when testing H0: β = β0 for fixed β0, when the instruments are weak and the variance matrix of the structural and first-stage errors is held constant. Power curves of the CLR and related tests have often been presented for a design where instead the variance matrix of the reduced-form and first-stage errors has been held constant. This latter design changes the endogeneity features at each value of β and results in a power curve that is close to the points with maximum power in the design with fixed vari- ance of the structural and first-stage errors. As the results for the design with fixed variance of the structural and first-stage errors are informative for the behaviour of the test-based confidence intervals, it seems more natural to consider this design. We find that LIML- and Fuller-based conditional Wald and conditional t0(βF ull)2 tests, which are not unbiased tests, are more powerful than the CLR test when the degree of endogeneity is low to moderate.