Residualised Treatment Intensity and the Estimation of Average Partial Effects

Abstract: This paper introduces R-OLS, an estimator for the average partial effect (APE) of a continuous treatment variable on an outcome variable in the presence of non-linear and non-additively separable confounding of unknown form. Identification of the APE is achieved by generalising Stein's Lemma (Stein, 1981), leveraging an exogenous error component in the treatment along with a flexible functional relationship between the treatment and the confounders. The identification results for R-OLS are used to characterize the properties of Double/Debiased Machine Learning (Chernozhukov et al., 2018), specifying the conditions under which the APE is estimated consistently. A novel decomposition of the ordinary least squares estimand provides intuition for these results. Monte Carlo simulations demonstrate that the proposed estimator outperforms existing methods, delivering accurate estimates of the true APE and exhibiting robustness to moderate violations of its underlying assumptions. The methodology is further illustrated through an empirical application to Fetzer (2019).