Fast convergence rates for dose-response estimation

Abstract

We consider the problem of estimating a dose-response curve. Continuous treatments arise often in practice, e.g. in the form of time spent on an operation, distance traveled to a location or dosage of a drug. Letting $A$ denote a continuous treatment variable, the target of inference is the expected outcome if everyone in the population takes treatment level $A=t$. Under standard assumptions, the dose-response function takes the form of a partial mean. Building upon the recent literature on nonparametric regression with estimated outcomes, our first contribution is to study global and local estimators of the dose-response based on empirical risk minimization. Our second and main contribution is to construct a $m^{\text{th}}$-order estimator based on the theory of higher-order influence functions. Under certain conditions, this higher order estimator achieves the fastest rate of convergence that we are aware of for this problem. However, the other two approaches are easier to implement using off-the-shelf software, since they are formulated as two-stage regression tasks. For each estimator, we provide an upper bound on the mean-square error and investigate its finite-sample performance through simulations and an empirical application. Finally, the supplementary material introduces a flexible, nonparametric approach for sensitivity analysis to violations of the no-unmeasured-confounding assumption with continuous treatments.