Summary: Power curves of the Conditional Likelihood Ratio \((CLR)\) and related tests for testing \(\mathrm{H}_0:\beta=\beta_0\) 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 \(|\beta|\), the fixed-\(\Omega\) design implicitly selects values for \(\Sigma\) where the power of the \(CLR\) test is high. We further show that the Likelihood Ratio statistic is identical to the \(t_0(\hat{\beta}_L)^2\) statistic as proposed by B. Mills et al. [J. Econom. 182, No. 2, 351–363 (2014; Zbl 1311.62099)], where \(\hat{\beta}_L\) 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 \(\rho_{u v}\) 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.