For a simplified structural equation/IV regression model with one right-side endogenous variable, we derive the exact conditional distribution function of Moreira’s (2003) conditional likelihood ratio (CLR) test statistic. This is used to obtain the critical value function needed to implement the CLR test, and reasonably comprehensive graphical versions of this function are provided for practical use. The analogous functions are also obtained for the case of testing more than one right-side endogenous coefficient, but in this case for a similar test motivated by, but not generally the same as, the likelihood ratio test. Next, the exact power functions of the CLR test, the Anderson-Rubin test, and the Lagrange multiplier test suggested by Kleibergen (2002) are derived and studied. The CLR test is shown to clearly conditionally dominate the other two tests for virtually all parameter configurations, but no test considered is either inadmissable or uniformly superior to the other two. The unconditional distribution function of the likelihood ratio test statistic is also derived using the same argument. This shows that both exactly, and under Staiger/Stock weak-instrument asymptotics, the test based on the usual asymptotic critical value is always oversized and can be very seriously so when the number of instruments is large.
Centre for microdata methods and practice