Ordinary least squares provides the optimal linear approximation to the true regression function. This paper investigates the Instrumental Variables (IV) version of this problem. The resulting parameter is called the Optimal Linear IV Approximation (OLIVA). The OLIVA is invariant to the distribution of the instruments. 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 IV model. The necessary regularity condition holds for a binary endogenous treatment, leading also to a LATE interpretation with positive weights in a fully heterogeneous model. The instrument in the IV estimand is unknown and may not be identified. A Two-Step IV (TSIV) estimator based on a Tikhonov regularized instrument 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 model. Monte Carlo simulations suggest a good finite sample performance for the proposed inferences  .
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 .
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This paper investigates the Instrumental Variables (IV) version of this problem. The resulting parameter
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The OLIVA is invariant to the distribution of the instruments.
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