Optimal Linear Instrumental Variables Approximations

TL;DR

This paper introduces a novel Two-Step IV estimator using Tikhonov regularization for optimal linear approximation of structural functions, enabling robust inference even without instrument identification.

Contribution

It proposes a new TSIV estimator that works under weak conditions, extends classical tests for exogeneity, and demonstrates strong finite sample performance.

Findings

TSIV estimator shows excellent finite sample properties.

Robust Hausman test remains valid under model misspecification.

Empirical application yields larger estimates of EIS, supporting exogeneity.

Abstract

This paper studies the identification and estimation of the optimal linear approximation of a structural regression function. The parameter in the linear approximation is called the Optimal Linear Instrumental Variables Approximation (OLIVA). This paper shows that a necessary condition for standard inference on the OLIVA is also sufficient for the existence of an IV estimand in a linear model. The instrument in the IV estimand is unknown and may not be identified. A Two-Step IV (TSIV) estimator based on Tikhonov regularization is proposed, which can be implemented by standard regression routines. We establish the asymptotic normality of the TSIV estimator assuming neither completeness nor identification of the instrument. As an important application of our analysis, we robustify the classical Hausman test for exogeneity against misspecification of the linear structural model. We also discuss extensions to weighted least squares criteria. Monte Carlo simulations suggest an excellent finite sample performance for the proposed inferences. Finally, in an empirical application estimating the elasticity of intertemporal substitution (EIS) with US data, we obtain TSIV estimates that are much larger than their standard IV counterparts, with our robust Hausman test failing to reject the null hypothesis of exogeneity of real interest rates.