Inference on Treatment Effects After Selection Amongst High-Dimensional Controls

TL;DR

This paper introduces a robust method for inferring treatment effects in high-dimensional settings with many controls, allowing for model misspecification and providing valid confidence intervals.

Contribution

It develops the post-double-selection method for uniform inference on treatment effects, accommodating imperfect control selection in high-dimensional models.

Findings

Method provides valid confidence intervals in high-dimensional models.

Applicable to Lasso and other sparse model selection techniques.

Demonstrated effectiveness through simulations and an application to crime rates.

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. Our analysis allows the number of controls to 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.