A unified framework for efficient estimation of general treatment models

Abstract

This paper presents a weighted optimization framework that unifies the binary, multivalued, and continuous treatment—as well as mixture of discrete and continuous treatment—under a unconfounded treatment assignment. With a general loss function, the framework includes the average, quantile, and asymmetric least squares causal effect of treatment as special cases. For this general framework, we first derive the semiparametric efficiency bound for the causal effect of treatment, extending the existing bound results to a wider class of models. We then propose a generalized optimization estimator for the causal effect with weights estimated by solving an expanding set of equations. Under some sufficient conditions, we establish the consistency and asymptotic normality of the proposed estimator of the causal effect and show that the estimator attains the semiparametric efficiency bound, thereby extending the existing literature on efficient estimation of causal effect to a wider class of applications. Finally, we discuss estimation of some causal effect functionals such as the treatment effect curve and the average outcome. To evaluate the finite sample performance of the proposed procedure, we conduct a small‐scale simulation study and find that the proposed estimation has practical value. In an empirical application, we detect a significant causal effect of political advertisements on campaign contributions in the binary treatment model, but not in the continuous treatment model.

A Unified Framework for Efficient Estimation of General Treatment Models

“…These high level conditions are nontrivial to verify. Most of our derivations are indeed verifying those high level conditions; see Section 4.2 of the Online Supplemental Material (Ai et al (2021)).…”

“…(2.2) (see Appendix A in the Replication file (Ai, Linton, Motegi, and Zhang (2021) for derivation), and hence the true value β * solves the weighted optimization problem…”

“…The proof of Theorem 1 is given in the Online Supplemental Material (Ai et al (2021), Section 2.1). It is worth noting that our bound V eff is equal to: the bound of Hahn (1998) for the case of binary average treatment, the bound of Cattaneo (2010) for the case of multivalued average treatment, and the bound of Firpo (2007) for the case of binary quantile treatment (see Ai et al (2021), Sections 2.2-2.4).…”

“…The proof of Theorem 1 is given in the Online Supplemental Material (Ai et al (2021), Section 2.1). It is worth noting that our bound V eff is equal to: the bound of Hahn (1998) for the case of binary average treatment, the bound of Cattaneo (2010) for the case of multivalued average treatment, and the bound of Firpo (2007) for the case of binary quantile treatment (see Ai et al (2021), Sections 2.2-2.4). Moreover, our bound applies to a much wider class of models, including quantile causal effect of multivalued, continuous, and mixture of discrete and continuous treatment as well as the asymmetric least squares estimation of the causal effect of all kinds of treatments.…”

“…Section 9 reports on a simulation study, while Section 10 presents an empirical application, followed by some concluding remarks in Section 11. All technical proofs and extra simulation results are relegated to the supplemental material (Ai et al (2021)).…”