Treatment Effect Models with Strategic Interaction in Treatment Decisions

Abstract

We develop identification and estimation methods for treatment effect models with strategic interaction in the treatment decisions. We consider models where one's treatment choice and outcome can be endogenously affected by others' treatment choices. We formulate the interaction of the treatment decisions as a two-player complete information game with potential multiple equilibria. For this model, using a latent index framework and the assumption of a stochastic equilibrium selection rule, we prove that the marginal treatment effect from one's own treatment and that from his/her partner's can be separately point-identified with potential over-identifiability. Based on our constructive identification results, we propose a two-step semiparametric procedure for estimating the marginal treatment effects using series approximation. The proposed estimator is shown to be uniformly consistent and asymptotically normal. As an empirical illustration, we investigate the impacts of risky behaviors on adolescents' academic performance.1 To the best of our knowledge, few studies address treatment evaluation in the presence of strategic interaction, where the treatment decisions are explicitly modeled as games. One important exception is Balat and Han (2018), who mainly investigate the partial identification of average treatment effects (ATE). Thus, our main parameters of interest (MTE) are different from theirs (ATE). In addition, the key identification assumptions in Balat and Han (2018) are shape restrictions on the conditional mean outcome functions, while we mainly rely on a stochastic equilibrium selection rule, as stated below.2 As another related work, Mogstad et al. (2019) introduce partial monotonicity to deal with issues due to multiple instruments. However, partial monotonicity is not designed for settings with multivalued or multidimensional treatments such as our situation.

Estimating Marginal Treatment Effects under Unobserved Group Heterogeneity

“…Thus, by (B.9), we obtain √ n m (1) i,K . Then, by the same argument as in the proof of Theorem 4.2 in Hoshino and Yanagi (2019), we obtain n i=1 E[φ 4 ji ] = O(ζ 2 0 (K)K/n) = o(1) under Assumption 3.4(ii). Hence, result (i) follows from Lyapunov's central limit theorem.…”

“…(iv), (v) The proofs are similar to the proof of Lemma A.3 in Hoshino and Yanagi (2019). For (iv), since min 1≤i≤n P ji > 0 for any j under Assumption 3.2, we have…”

“…(v), (vi) The proofs are similar to the proof of Lemma A.3 inHoshino and Yanagi (2019). For (vi), notethat E[e (1) |D, X, Z] = E[E[e (1) |D, X, Z, s]|D, X, Z] = 0 since E[e (1) |D, X, Z, s = j] = E…”

A pairwise strategic network formation model with group heterogeneity: With an application to international travel

“…We provide an explicit form of in Appendix C.1. For another identification condition for , see, for example, Theorem B.2 of Hoshino & Yanagi (2021), that is based on the monotonicity of the likelihood function with respect to .…”

Estimating a Continuous Treatment Model with Spillovers: A Control Function Approach

“…It is easy to see that ζ : gives the order of ||BP U V pu, vq{Bu|| and ||BP U V pu, vq{Bv||. For example, when one uses a tensor product B-splines, it can be shown that ζ : " Opκ n q (see, e.g., Hoshino and Yanagi, 2021b). Condition (ii) implies the first part of (i).…”