Estimating endogenous treatments effects under long-range dependency without untreated controls

The identification and estimation of social policy effects through time‑series natural experiments is fundamental in modern econometrics. However, challenges come from the heterogeneities caused by staggered treatment adoptions and the endogeneities caused by omitted variables. In this paper, we propose a novel method to identify and estimate staggeringly adopted endogenous treatments effects with several treatments when there is no available (suitable) untreated unit and instrument variable. First, we propose a conditional mean symmetry condition by projecting the potential outcomes onto a sub-linear space spanned by the proposed common proximal variable. Under this condition, we can rule out confounding biases. Second, a proposed weak index restriction constructed by Bernstein expansions satisfying conditional mean independence property enables us to consistently estimate multiple heterogeneous treatments effects, and the proposed estimators are robust to weak common proximal variable. We show that the asymptotic distribution of the step-wise estimator is a fractional Brownian motion process with long range dependency. Third, we propose a bootstrap procedure to circumvent the inference difficulty brought by time-series dependencies. Monte Carlo simulations show that our proposed estimator and inference framework work well in small samples, and our contributions are further illustrated by an empirical example with unilateral divorce law reforms.