# CFDP 2073R

### On Optimal Inference in the Linear IV Model

**Author(s):**

**Publication Date: **January 2017

**Revision Date: **March 2018

**Pages: **34

**Abstract: **

This paper considers tests and conﬁdence sets (CS’s) 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 .005 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 (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, theory and numerical results indicate 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. Here, “close” means .015 or less for k≤20, where k denotes the number of IV’s, and .025 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 CS’s have very good overall properties.

**Supplemental Material: **Supplemental material

**Supplement pages: **77

**Keywords: **

Conditional likelihood ratio test, Conﬁdence interval, Inﬁnite length, Linear instrumental variables, Optimal test, Weighted average power, Similar test