Econometrica: Nov 2012, Volume 80, Issue 6
Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain
A. Belloni, D. Chen, V. Chernozhukov, C. HansenWe develop results for the use of Lasso and post‐Lasso methods to form first‐stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, . Our results apply even when is much larger than the sample size, . We show that the IV estimator based on using Lasso or post‐Lasso in the first stage is root‐ consistent and asymptotically normal when the first stage is approximately sparse, that is, when the conditional expectation of the endogenous variables given the instruments can be well‐approximated by a relatively small set of variables whose identities may be unknown. We also show that the estimator is semiparametrically efficient when the structural error is homoscedastic. Notably, our results allow for imperfect model selection, and do not rely upon the unrealistic “beta‐min” conditions that are widely used to establish validity of inference following model selection (see also Belloni, Chernozhukov, and Hansen (2011b)). In simulation experiments, the Lasso‐based IV estimator with a data‐driven penalty performs well compared to recently advocated many‐instrument robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the Lasso‐based IV estimator outperforms an intuitive benchmark.
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