This paper studies the identi.cation 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 su¢ cient for the existence of an IV estimand in a linear model. The instrument in the IV estimand is unknown and may not be identi.ed. 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 identi.cation of the instrument. As an important application of our analysis, we robustify the classical Hausman test for exogeneity against misspeci.cation of the linear structural model. We also discuss extensions to weighted least squares criteria. Monte Carlo simulations suggest an excellent .nite 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.

Juan Carlos Escanciano  is  Research Chair in Economics and Full Professor at Universidad Carlos III de Madrid. He obtained his P.hD in Economics at Universidad Carlos III de Madrid in 2004.  He obtained an Assistant Professor position at Universidad de Navarra (2004-2006), Full Professor (with tenure) at Indiana University (2006-2018), and Visiting positions at Yale University, Cornell, Rochester and MIT. His research and teaching focus on Econometric Theory, including identification, estimation and specification testing, as well as empirical work in Financial Econometrics and Risk Management.

Juan Carlos is Fellow of Journal of Econometrics. He has published papers in several leading international journals, including Journal of American Statistical Association, Journal of Econometrics, Econometric Theory, Quantitative Economics, Management Science, and The Annals of Statistics. He is Associate Editor of  Series, Econometric Theory, Econometric Reviews, Journal of Business and Economic Statistics, and Co-Editor senior in Advances in Econometrics.