Difference-in-differences with stochastic policy shifts of a continuous treatment

Abstract

Treatment effects of stochastic policy shifts quantify differences in outcomes across counterfactual scenarios with varying treatment distributions. Stochastic policy shifts may be of interest in settings where it is unrealistic or infeasible to deterministically manipulate treatments. In this paper, methods are developed to draw inference about stochastic policy effects under difference-in-differences (DiD) designs with a continuous treatment. The proposed causal estimand is the expected effect of modifying the continuous dose distribution among the treated, i.e., those that received a non-zero dose. Several possible stochastic policies are discussed and a general framework for identification and estimation is proposed. One stochastic policy applicable to many settings is the exponential tilt, which increments the conditional density function of the continuous dose. For the exponential tilt policy, a double/debiased machine learning estimator is proposed that allows for data-adaptive, nonparametric nuisance function estimation. Under mild convergence rate conditions, the estimator is shown to be root-$n$ consistent and asymptotically normal with variance attaining the nonparametric efficiency bound. The proposed method is used to study the effect of hydraulic fracturing activity on employment and income.