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Simultaneous inference for Best Linear Predictor of the Conditional Average Treatment Effect and other structural functions

Authors: Victor Chernozhukov and Vira Semenova
Date: 04 July 2018
Type: cemmap Working Paper, CWP40/18

Abstract

This paper provides estimation and inference methods for a structural function, such as Conditional Average Treatment Effect (CATE), based on modern machine learning (ML) tools. We assume that such function can be represented as a conditional expectation  = of a signal , where is the unknown nuisance function. In addition to CATE, examples of such functions include regression function with Partially Missing Outcome and Conditional Average Partial Derivative. We approximate  by a linear form , where  is a vector of the approximating functions and  is the Best Linear Predictor. Plugging in the fi rst-stage estimate  into the signal , we estimate  via ordinary least squares of  on . We deliver a high-quality estimate  of the pseudo-target function , that features (a) a pointwise Gaussian approximation of  at a point , (b) a simultaneous Gaussian approximation of  uniformly over x, and (c) optimal rate of convergence of to  uniformly over x. In the case the misspeci cation error of the linear form decays sufficiently fast, these approximations automatically hold for the target function  instead of a pseudo-target . The fi rst stage nuisance parameter  is allowed to be high-dimensional and is estimated by modern ML tools, such as neural networks, -shrinkage estimators, and random forest. Using our method, we estimate the average price elasticity conditional on income using Yatchew and No (2001) data and provide uniform con fidence bands for the target regression function.

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