On the power of the conditional likelihood ratio and related tests for weak-instrument robust inference

Power curves of the Conditional Likelihood Ratio (CLR) and related tests for testing H0:β = β0 in linear models with a single endogenous variable, y = xβ+u, estimated using potentially weak instrumental variables have been presented for two different designs. One design keeps the variance matrix of the structural and first-stage errors, Σ, constant, the other instead keeps the variance matrix of the reduced-form and first-stage errors, Ω, constant. The values of Σ govern the endogeneity features of the model. The fixed-Ω design changes these endogeneity features with changing values of β in a way that makes it less suitable for an analysis of the behaviour of the tests in low to moderate endogeneity settings, or when β and the correlation of the structural and first-stage errors, ρuv, have the same sign. At larger values of |β|, the fixed-Ω design implicitly selects values for Σ where the power of the CLR test is high. We show that the Likelihood Ratio statistic is identical to the t0(βb L) 2 statistic as proposed by Mills, Moreira, and Vilela (2014), where βb L is the LIML estimator. In fixed-Σ design Monte Carlo simulations, we find that LIMLand Fuller-based conditional Wald tests and the Fuller-based conditional t 20 test are more powerful than the CLR test when the degree of endogeneity is low to moderate. The conditional Wald tests are further the most powerful of these tests when β and ρuv have the same sign. We show that in the fixed-Ω design, setting β0 = 0 and the diagonal elements of Ω equal to 1 is not without loss of generality, unlike in the fixed-Σ design.