Double Debiased Machine Learning Nonparametric Inference with Continuous Treatments

Abstract

We propose a nonparametric inference method for causal effects of continuous treatment variables, under unconfoundedness and in the presence of high-dimensional or nonparametric nuisance parameters. Our double debiased machine learning (DML) estimators for the average doseresponse function (or the average structural function) and the partial effects are asymptotically normal with nonparametric convergence rates. The nuisance estimators for the conditional expectation function and the conditional density can be nonparametric kernel or series estimators or ML methods. Using a kernel-based doubly robust influence function and cross-fitting, we give tractable primitive conditions under which the nuisance estimators do not affect the first-order large sample distribution of the DML estimators. We justify the use of kernel to localize the continuous treatment at a given value by the Gateaux derivative. We implement various ML methods in Monte Carlo simulations and an empirical application on a job training program evaluation.

Kernel Methods for Multistage Causal Inference: Mediation Analysis and Dynamic Treatment Effects

“…Theorem 8 (Semiparametic consistency, Gaussian approximation, and efficiency of dynamic treatment effects) Suppose the conditions of Theorem 7 hold, as well as Assumptions 12,13,14,and 15. Set…”

“…Next, we propose a dynamic dose response design, extending a static dose response design [15] to the multistage setting. The nonlinear counterfactual function is…”

“…Chronologically, X 1 ∈ 65 are covariates at baseline; D 1 ∈ is total hours spend in classes in the first year after randomization; X 2 ∈ 30 are covariates observed at the end of the first year after randomization; D 2 ∈ is total hours spend in classes in the second year after randomization; and Y ∈ is proportion of weeks employed in the fourth year after randomization. 9 We use similar covariates and sample as [15], with n = 3, 141.…”

“…At best, the effect of receiving job training is 64% employment, compared to receiving no class-hours at all which gives 56% employment. This maximum effect is achieved by 480-1280 class-hours (12-32 weeks) in year one and 0-480 (0-12 9 By contrast, static dose response analysis of this dataset takes X = X1, D = D1, and Y ∈ as the fraction of weeks employed in the second year after randomization [15,65]. 10 X1 ∈ 65 yet X2 ∈ 30 for two reasons.…”

A Finite Sample Theorem for Longitudinal Causal Inference with Machine Learning: Long Term, Dynamic, and Mediated Effects

“…Despite the additional complexity of the longitudinal setting, I recover equally strong results. In particular, [54,52,14,28,101,35,31,51,29,2] study static local parameters and [81,50,40,29] study static proximal parameters. Like the latter, I arrive at mean square rates and projected mean square rates of confounding bridges (also called nonparametric instrumental variable regressions).…”