Power Bounds and Efficiency Loss for Asymptotically Optimal Tests in IV Regression

TL;DR

This paper characterizes the limits of power and size in overidentified IV tests with complex errors, showing that certain tests can nearly maximize power but may be inefficient, with implications demonstrated through empirical data.

Contribution

It introduces a decision theoretic framework for understanding power bounds in IV tests with heteroskedastic or autocorrelated errors, highlighting the efficiency loss of common tests.

Findings

CLR test attains the decision frontier when the problem is non-trivial.

Lagrange multiplier and quasi likelihood ratio tests can have power close to size.

Empirical analysis shows these issues are relevant in practice.

Abstract

We characterize the maximal attainable power-size gap in overidentified instrumental variables models with heteroskedastic or autocorrelated (HAC) errors. Using total variation distance and Kraft's theorem, we define the decision theoretic frontier of the testing problem. We show that Lagrange multiplier and conditional quasi likelihood ratio tests can have power arbitrarily close to size even when the null and alternative are well separated, because they do not fully exploit the reduced-form likelihood. In contrast, the conditional likelihood ratio (CLR) test uses the full reduced-form likelihood. We prove that the power-size gap of CLR converges to one if and only if the testing problem becomes trivial in total variation distance, so that CLR attains the decision theoretic frontier whenever any test can. An empirical illustration based on Yogo (2004) shows that these failures arise in empirically relevant configurations.