Applicability of semi-parametric procedures for Conditional Parallel Shifts with continuous covariates

Establish whether semi-parametric or related estimation and inference procedures developed for difference-in-differences with continuous covariates (such as those in Chang, 2020, and Sant’Anna and Zhao, 2020) can be applied to the Conditional Parallel Shifts framework for interval-valued outcomes; in particular, demonstrate that such procedures yield valid estimation and inference for the endpoints of the sharp identified set Θ_CPS for the average treatment effect on the treated when covariates X_i are continuously distributed under Assumption Conditional-PS.

Background

The paper extends difference-in-differences (DID) methodology to interval-valued outcomes and proposes the Parallel Shifts assumption to identify bounds on the average treatment effect on the treated (ATT). In the presence of covariates, the authors introduce the Conditional Parallel Shifts (CPS) framework, which provides sharp identified sets for the ATT by mapping the control group’s interval means to the treated group’s interval means conditional on covariates.

For discrete covariates, the authors indicate that their inference approach based on Imbens and Manski (2004) and Stoye (2009) carries over straightforwardly. However, when covariates are continuously distributed, they note that some form of (semi-)parametric modeling is necessary, referencing existing DID methods for continuous covariates. They explicitly conjecture that similar procedures should be applicable to their CPS setting but do not provide a formal proof, leaving the question open.