Double Machine Learning Density Estimation for Local Treatment Effects with Instruments

Randomized Controlled Trials constitute a powerful tool to learn cause and effect relationships found throughout a wide range of applied settings. In practice, the treatment assignment's compliance is hard to ascertain in many settings since patients may not feel compelled to take the treatment for various reasons. One typical quantity investigated in these settings is the local treatment effect (LTE, for short). The LTE measures the causal effect among compliers, which usually comes under the assumption of monotonicity (only the ones offered the treatment are allowed to take it). In this paper, we investigate the challenge of estimating the LTE density function (instead of its expected value) of a binary treatment on a continuous outcome given a binary instrumental variable in the presence of both observed and unobserved confounders. Specifically, we develop two families of methods for this task, kernel-smoothing and model-based approximationsthe former smoothes the density by convoluting with a smooth kernel function; the latter projects the density onto a finite-dimensional density class. For both approaches, we derive double/debiased machine learning (DML) based estimators. We study the asymptotic convergence rates of the estimators and show that they are robust to the biases in nuisance function estimation. We illustrate the proposed methods on synthetic data and a real dataset called 401(k).