Estimation of Treatment Effects under Essential Heterogeneity

Assume that the economic model generating the data for potential outcomes is of the form: Y1 = α1 + φ+Xβ1 + U1 (1) Y0 = α0 +Xβ0 + U0 (2) where vector X represents the observed variables (regressors), U0 and U1 represent the unobservables in the potential outcome equations, and φ represents the benefit associated with the treatment (D = 1). The assumptions of linearity and additive separability are not intrinsic to the model of essential heterogeneity and are used in this supplement just to illustrate the estimation method we propose. The individuals decide whether or not to receive the treatment (D = 1 or D = 0) based on a latent variable I I = γ0Z − V (3) where Z and V represent observables and unobservables respectively. Thus, we can define a binary variable D indicating the treatment status D = 1⁄2 1 if I > 0 0 if I ≤ 0 (4)