Ordinary least squares provides the optimal linear approximation to the true regression function under misspecification . This paper investigates the Instrumental Variables (IV) version of this problem. The resulting population parameter is called the Optimal Linear IV Approximation (OLIVA).  This paper shows that a necessary condition for regular identification of the OLIVA is also sufficient for  existence of an IV estimand in a linear IV model. The necessary  condition holds for the important case of a binary endogenous treatment, leading also to a LATE interpretation with positive weights  . The instrument in the IV estimand is unknown and is estimated in a first step . A Two-Step IV (TSIV) estimator is proposed . We establish the asymptotic normality of a debiased TSIV estimator based on locally robust moments. The TSIV estimator does not require neither completeness nor identification of the instrument. As a by-product of our analysis, we robustify the classical Hausman test for exogeneity against misspecification of the linear model. Monte Carlo simulations suggest excellent finite sample performance for the proposed inferences.
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.
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under misspecification
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population
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The OLIVA is invariant to the distribution of the instruments.
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regular identification of
standard inference on
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the
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regularity
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the important case of
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in a fully heterogeneous model
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is estimated in a first step
may not be identified
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is proposed
based on a Tikhonov regularized instrument is proposed, which can be implemented by standard regression routines
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a debiased TSIV estimator based on locally robust moments. The TSIV estimator does not require
the TSIV estimator assuming
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a by-product
an important application
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excellent
a good