Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain

Posted: 5 Jun 2017

See all articles by Alexandre Belloni

Alexandre Belloni

Duke University - Fuqua School of Business

Daniel L. Chen

Directeur de Recherche, Centre National de la Recherche Scientifique, Toulouse School of Economics, Institute for Advanced Study in Toulouse, University of Toulouse Capitole, Toulouse, France

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics; New Economic School

Christian Hansen

University of Chicago - Booth School of Business

Date Written: Novemeber 2012

Abstract

We 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, p. Our results apply even when p is much larger than the sample size, n. We show that the IV estimator based on using Lasso or post-Lasso in the first stage is root-n 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 Lassobased IV estimator outperforms an intuitive benchmark.

Optimal instruments are conditional expectations. In developing the IV results, we establish a series of new results for Lasso and post-Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non-Gaussian, heteroscedastic disturbances that uses a data-weighted ℓ1-penalty function. By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for the resulting Lasso and post-Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that log p = o (n1/3). We also provide a data-driven method for choosing the penalty level that must be specified in obtaining Lasso and post-Lasso estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.

Keywords: Inference on a Low-Dimensional Parameter after Model Selection, Imperfect Model Selection, Instrumental Variables, Lasso, Post-Lasso, Data-Driven Penalty, Heteroscedasticity, Non-Gaussian Errors, Moderate Deviations for Self-Normalized Sums

Suggested Citation

Belloni, Alexandre and Chen, Daniel L. and Chernozhukov, Victor and Hansen, Christian, Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain (Novemeber 2012). Econometrica, Vol. 80, No. 6, p. 2369–2429, November 2012 . Available at SSRN: https://ssrn.com/abstract=2975719

Alexandre Belloni

Duke University - Fuqua School of Business ( email )

Box 90120
Durham, NC 27708-0120
United States

Daniel L. Chen (Contact Author)

Directeur de Recherche, Centre National de la Recherche Scientifique, Toulouse School of Economics, Institute for Advanced Study in Toulouse, University of Toulouse Capitole, Toulouse, France ( email )

21 allée de Brienne
31015 Toulouse cedex 6 France
Toulouse, 31015
France

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics ( email )

50 Memorial Drive
Room E52-262f
Cambridge, MA 02142
United States
617-253-4767 (Phone)
617-253-1330 (Fax)

HOME PAGE: http://www.mit.edu/~vchern/

New Economic School

100A Novaya Street
Moscow, Skolkovo 143026
Russia

Christian Hansen

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

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