A Note on Optimal Inference in the Linear IV Model

This paper considers tests and con…dence sets (CS’s) concerning the coe¢ cient 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 …nite-sample lower bound function for the probability that a CS constructed using a two-sided invariant similar test has in…nite length and shows numerically that the conditional likelihood ratio (CLR) CS of Moreira (2003) is not always “very close”to this lower bound function. This implies that the CLR test is not always very close to the two-sided asymptotically-e¢ cient (AE) power envelope for invariant similar tests of Andrews, Moreira, and Stock (2006) (AMS). On the other hand, the paper establishes the …nite-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 (non-zero) 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, the paper shows via theory and numerical work that the CLR test is close to having greatest average power, where the average is over a grid of concentration parameter values and over pairs alternative hypothesis values of the parameter of interest, uniformly over pairs of alternative hypothesis values and uniformly over the correlation between the structural and reduced-form errors. 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 CS’s have very good overall properties. Keywords: Conditional likelihood ratio test, con…dence interval, in…nite length, linear instrumental variables, optimal test, weighted average power, similar test. JEL Classi…cation Numbers: C12, C36.