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 $H_{0}\beta=\beta_{0}$ in linear models with a single endogenous variable, $y=x\beta+u$, 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, $\Sigma$, constant, the other instead keeps the variance matrix of the reduced-form and first-stage errors, $\Omega$, constant. The values of $\Sigma$ govern the endogeneity features of the model. The fixed-$\Omega$ design changes these endogeneity features with changing values of $\beta$ 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 $\beta$ and the correlation of the structural and first-stage errors, $\rho_{uv}$, have the same sign. At larger values of $\left|\beta\right|$, the fixed-$\Omega$ design implicitly selects values for $\Sigma$ where the power of the $CLR$ test is high. We show that the Likelihood Ratio statistic is identical to the $t_{0}(\widehat{\beta}_{L})^{2}$ statistic as proposed by Mills, Moreira and Vilela (2014), where $\widehat{\beta}_{L}$ is the LIML estimator. In fixed-$\Sigma$ design Monte Carlo simulations, we find that LIML- and Fuller-based conditional Wald tests and the Fuller-based conditional $t_{0}^{2}$ 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 $\beta$ and $\rho_{uv}$ have the same sign. We show that in the fixed-$\Omega$ design, setting $\beta_{0}=0$ and the diagonal elements of $\Omega$ equal to $1$ is not without loss of generality, unlike in the fixed-$\Sigma$ design. JEL codes: C12, C26