# Econometrica

## Journal Of The Econometric Society

### An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: May, 2006, Volume 74, Issue 3

# Optimal Two‐Sided Invariant Similar Tests for Instrumental Variables Regression

Donald W. K. Andrews, Marcelo J. Moreira, James H. Stock

This paper considers tests of the parameter on an endogenous variable in an instrumental variables regression model. The focus is on determining tests that have some optimal power properties. We start by considering a model with normally distributed errors and known error covariance matrix. We consider tests that are similar and satisfy a natural rotational invariance condition. We determine a two‐sided power envelope for invariant similar tests. This allows us to assess and compare the power properties of tests such as the conditional likelihood ratio (CLR), the Lagrange multiplier, and the Anderson–Rubin tests. We find that the CLR test is quite close to being uniformly most powerful invariant among a class of two‐sided tests.

## Supplemental Material

### Supplement to "Optimal Two-sided Invariant Similar Tests for Instrumental Variables Regression"

This paper contains supplemental material to Andrews, Moreira, and Stock (2006a), hereafter AMS. Section 2 provides details concerning the sign-invariant power envelope for similar tests introduced in AMS. Section 3 does likewise for the LU power envelope for invariant similar tests. Section 4 reports additional numerical results to those in AMS. Section 5 establishes consistency of the covariance matrix estimator in AMS. Section 6 gives proofs of Lemmas 1 and 2 of AMS. Section 7 proves the claim made in Comment 2 to Corollary 1 of AMS that when k =1 the optimal invariant similar test in terms of two-point weighted average power is the Anderson-Rubin test (which is equivalent in this case to the LM and CLR tests). An Appendix describes numerical methods used in Section 4.

### Supplement to "Optimal Two-sided Invariant Similar Tests for Instrumental Variables

This file contains tables of the conditional critical values of the CLR test presented in Andrews, Moreira, and Stock (2006a).

### Supplement to "Optimal Two-sided Invariant Similar Tests for Instrumental Variables Regression"

This paper contains supplemental material to Andrews, Moreira, and Stock (2006a), hereafter AMS. Section 2 provides details concerning the sign-invariant power envelope for similar tests introduced in AMS. Section 3 does likewise for the LU power envelope for invariant similar tests. Section 4 reports additional numerical results to those in AMS. Section 5 establishes consistency of the covariance matrix estimator in AMS. Section 6 gives proofs of Lemmas 1 and 2 of AMS. Section 7 proves the claim made in Comment 2 to Corollary 1 of AMS that when k =1 the optimal invariant similar test in terms of two-point weighted average power is the Anderson-Rubin test (which is equivalent in this case to the LM and CLR tests). An Appendix describes numerical methods used in Section 4.

### Supplement to "Optimal Two-sided Invariant Similar Tests for Instrumental Variables

This file contains tables of the conditional critical values of the CLR test presented in Andrews, Moreira, and Stock (2006a).