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 further show that the Likelihood Ratio statistic is identical to the t0(βˆL)2 statistic as proposed by Mills et al. (2014), where βˆL is the Liml estimator. In fixed-Σ design Monte Carlo simulations, we find that Liml- and Fuller-based conditional Wald tests and the Fuller-based conditional t02 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.