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Inference on treatment effects after selection amongst high-dimensional controls

Authors: Alexandre Belloni , Victor Chernozhukov and Christian Hansen
Date: 07 May 2012
Type: cemmap Working Paper, CWP10/12
DOI: 10.1920/wp.cem.2012.1012


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.

This paper is a revision of CWP42/11.

Download full version
New version:
Alexandre Belloni, Victor Chernozhukov and Christian Hansen July 2013, Inference on treatment effects after selection amongst high-dimensional controls, cemmap Working Paper, cemmap
Previous version:
Alexandre Belloni, Victor Chernozhukov and Christian Hansen December 2011, Estimation of treatment effects with high-dimensional controls, cemmap Working Paper

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