Double Machine Learning Density Estimation for Local Treatment Effects with Instruments

Local treatment effects are a common quantity found throughout the empirical sciences that measure the treatment effect among those who comply with what they are assigned. Most of the literature is focused on estimating the average of such quantity, which is called the ``local average treatment effect (LATE)'' [Imbens and Angrist, 1994]). In this work, we study how to estimate the density of the local treatment effect, which is naturally more informative than its average. Specifically, we develop two families of methods for this task, namely, kernel-smoothing and model-based approaches. The kernel-smoothing-based approach estimates the density through some smooth kernel functions. The model-based approach estimates the density by projecting it onto a finite-dimensional density class. For both approaches, we derive the corresponding double/debiased machine learning-based estimators [Chernozhukov et al., 2018]. We further study the asymptotic convergence rates of the estimators and show that they are robust to the biases in nuisance function estimation. The use of the proposed methods is illustrated through both synthetic and a real dataset called 401(k).