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

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 g(x) = E[Yη 0 |X = x] of a signal Yη 0 , where η0 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 g(x) by a linear form p(x) β0, where p(x) is a vector of the approximating functions and β0 is the Best Linear Predictor. Plugging in the first-stage estimate η into the signal Y η , we estimate β0 via ordinary least squares of Y η on p(X). We deliver a high-quality estimate p(x) β of the pseudo-target function p(x) β0, that features (a) a pointwise Gaussian approximation of p(x0) β0 at a point x0, (b) a simultaneous Gaussian approximation of p(x) β0 uniformly over x, and (c) optimal rate of convergence of p(x) β to p(x) β0 uniformly over x. In the case the misspecification error of the linear form decays sufficiently fast, these approximations automatically hold for the target function g(x) instead of a pseudo-target p(x) β0. The first stage nuisance parameter η0 is allowed to be high-dimensional and is estimated by modern ML tools, such as neural networks, l1-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 confidence bands for the target regression function.

Robust Machine Learning for Treatment Effects in Multilevel Observational Studies Under Cluster-level Unmeasured Confounding

“…In the ECLS-K study, τ would be the overall effect of taking an algebra course in eighth grade on math achievement scores. The second causal estimand is a type of conditional average treatment effect (CATE) known as the best linear projection of the CATE (Ding, Feller, & Miratrix, 2019;Semenova & Chernozhukov, 2020). To motivate this estimand, consider the usual CATE which measures the ATE among a subgroup of individuals defined by p v pre-treatment covariates…”

“…To rely less on such functional assumptions, Semenova and Chernozhukov (2020) proposed to estimate the best linear projection of the CATE, or formally:…”

“…In particular, the estimator τ is consistent for the ATE as long as the propensity score model or the outcome model is correctly specified, but not necessarily both. Additionally, following Semenova and Chernozhukov (2020), we can use the doubly robust estimator to estimate the best linear approximation coefficients β by running a linear regression of τij on V ij :…”

“…For standard errors, following Semenova and Chernozhukov (2020), we use the standard errors from ordinary least squares (OLS) regression where we treat τ ij as the outcome variable and V ij as the predictors and extract the OLS standard errors for the regression coefficients β associated with V ij .…”

Robust Machine Learning for Treatment Effects in Multilevel Observational Studies Under Cluster-level Unmeasured Confounding

“…In the ECLS-K study, τ would be the overall effect of taking an algebra course in eighth grade on math achievement scores. The second causal estimand is a type of conditional average treatment effect (CATE) known as the best linear projection of the CATE (Carvalho et al, 2019;Semenova & Chernozhukov, 2020) and to motivate this estimand, consider the usual CATE which measures the ATE among a subgroup of individuals defined by…”