Efficiency Loss of Asymptotically Efficient Tests in an Instrumental Variables Regression

TL;DR

This paper investigates the limitations of certain statistical tests in instrumental variable models, revealing conditions under which these tests lose power and applying the theory to real-world economic data.

Contribution

It identifies conditions causing power loss in instrumental variable tests and demonstrates their practical relevance in economic inference on intertemporal elasticity.

Findings

Lagrange Multiplier test can have power close to size under certain constraints.

Severe power losses occur in cases with eigenvalues of opposite signs in the covariance matrix.

Most studied countries exhibit these power loss conditions at the 95% confidence level.

Abstract

In an instrumental variable model, the score statistic can be bounded for any alternative in parts of the parameter space. These regions involve a constraint on the first-stage regression coefficients and the reduced-form covariance matrix. Consequently, the Lagrange Multiplier test can have power close to size, despite being efficient under standard asymptotics. This information loss limits the power of conditional tests which use only the Anderson-Rubin and the score statistic. The conditional quasi-likelihood ratio test also suffers severe losses because it can be bounded for any alternative. A necessary condition for drastic power loss to occur is that the Hermitian of the reduced-form covariance matrix has eigenvalues of opposite signs. These cases are denoted impossibility designs (ID). We show this happens in practice, by applying our theory to the problem of inference on the intertemporal elasticity of substitution (IES). Of eleven countries studied by Yogo (2004} and Andrews (2016), nine are consistent with ID at the 95\% level.