Optimal Linear Instrumental Variables Approximations

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.