Inference on Treatment Effects after Selection Amongst High-Dimensional Controls

67 Pages Posted: 5 May 2012  

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center

Victor Chernozhukov

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

Christian Hansen

University of Chicago - Booth School of Business - Econometrics and Statistics

Date Written: May 3, 2012

Abstract

We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances where the number of controls may be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the “post-double-selection” method. Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties.

The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We illustrate the use of the developed methods with numerical simulations and an application to the effect of abortion on crime rates.

Keywords: treatment effects, partially linear model, high-dimensional-sparse regression, inference under imperfect model selection, uniformly valid inference after model selection

JEL Classification: C10, C51

Suggested Citation

Belloni, Alexandre and Chernozhukov, Victor and Hansen, Christian, Inference on Treatment Effects after Selection Amongst High-Dimensional Controls (May 3, 2012). MIT Department of Economics Working Paper No. 12-13. Available at SSRN: https://ssrn.com/abstract=2051129 or http://dx.doi.org/10.2139/ssrn.2051129

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center ( email )

77 Massachusetts Avenue
Bldg. E 40-149
Cambridge, MA 02139
United States

Victor Chernozhukov (Contact Author)

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

47 Nakhimovsky Prospekt
Moscow, 117418
Russia

Christian Hansen

University of Chicago - Booth School of Business - Econometrics and Statistics ( email )

Chicago, IL 60637
United States
773-834-1702 (Phone)

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