On optimal inference in the linear IV model

This paper considers tests and confidence sets (CSs) concerning the coefficient on the endogenous variable in the linear IV regression model with homoskedastic normal errors and one right-hand side endogenous variable. The paper derives a finite-sample lower bound function for the probability that a CS constructed using a two-sided invariant similar test has infinite length and shows numerically that the conditional likelihood ratio (CLR) CS of Moreira (2003) is not always “very close,” say 0005 or less, to this lower bound function. This implies that the CLR test is not always very close to the two-sided asymptotically-efficient (AE) power envelope for invariant similar tests of Andrews, Moreira, and Stock (2006) (AMS). On the other hand, the paper establishes the finite-sample optimality of the CLR test when the correlation between the structural and reduced-form errors, or between the two reduced-form errors, goes to 1 or -1 and other parameters are held constant, where optimality means achievement of the two-sided AE power envelope of AMS. These results cover the full range of (nonzero) IV strength. The paper investigates in detail scenarios in which the CLR test is not on the two-sided AE power envelope of AMS. Also, theory and numerical results indicate that the CLR test is close to having the greatest average power, where the average is over a specified grid of concentration parameter values and over a pair of alternative hypothesis values of the parameter of interest, uniformly over all such pairs of alternative hypothesis values and uniformly over the correlation between the structural and reduced-form errors. Here, “close” means 0015 or less for k = 20, where k denotes the number of IVs, and 0025 or less for 0 < k = 40. The paper concludes that, although the CLR test is not always very close to the two-sided AE power envelope of AMS, CLR tests and CSs have very good overall properties.