Double Debiased Machine Learning Nonparametric Inference with Continuous Treatments

We propose a doubly robust inference method for causal effects of continuous treatment variables, under unconfoundedness and with nonparametric or high-dimensional nuisance functions. Our double debiased machine learning (DML) estimators for the average dose-response function (or the average structural function) and the partial effects are asymptotically normal with nonparametric convergence rates. The first-step estimators for the nuisance conditional expectation function and the conditional density can be nonparametric or ML methods. Utilizing a kernel-based doubly robust moment function and cross-fitting, we give high-level conditions under which the nuisance function estimators do not affect the first-order large sample distribution of the DML estimators. We provide sufficient low-level conditions for kernel, series, and deep neural networks. 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.

我们提出了一种针对连续处理变量因果效应的双重稳健推断方法，适用于无混杂性假设下的非参数或高维扰动函数场景。针对平均剂量响应函数（average dose-response function）与平均结构函数（average structural function）以及偏效应，我们提出的双重去偏机器学习（double debiased machine learning (DML)）估计量渐近服从正态分布且具备非参数收敛速率。用于估计扰动条件期望函数与条件密度的第一步估计量可采用非参数方法或机器学习（ML）方法。我们利用基于核的双重稳健矩函数与交叉拟合方法，给出了高阶条件，在此条件下扰动函数估计量不会对DML估计量的一阶大样本分布产生影响。我们为核方法、级数法与深度神经网络提供了充分的低阶条件。我们通过加托导数（Gateaux derivative）验证了使用核方法对给定取值处的连续处理进行局部化的合理性。我们在蒙特卡洛模拟（Monte Carlo simulations）以及一项职业培训项目评估的实证应用中实现了多种机器学习方法。