Variance-Weighted Effect of Endogenous Treatment and the Estimand of Fixed-Effect Approach

Given an endogenous binary treatment D, an outcome Y and covariates Z, finding an instrument for D is far from easy. Instead, this paper deals with the endogeneity using two-wave (t=1,2) panel data, assuming that the endogeneity is caused by a time-constant error δ_{i}. We postulate that Y_{it} is generated by a semiparametric model with an unknown heterogeneous treatment effect μ_{D}(Z_{it}) where δ_{i} appears additively, so that δ_{i} drops out for ΔY_{i}≡Y_{i2}-Y_{i1}. The main difficulty with ΔY_{i} is that the resulting effect takes a differenced form Δμ_{D}(Z_{it}), not an additive form of μ_{D}(Z_{i1}) and μ_{D}(Z_{i2}). Despite this difficulty, however, a "variance- (or overlap-) weighted" average of μ_{D}(Z_{i1}) and μ_{D}(Z_{i2}) is estimated with the ordinary least squares estimator (OLS) of ΔY_{i} on the difference of the `propensity score residual', without a direct nonparametric estimation of μ_{D}(Z_{it}). Also, this finding answers an important practical question: what is estimated by the popular `fixed-effect/within-group' estimator for panel constant-effect linear models when the effect is actually not a constant? The answer is essentially the variance-weighted average of μ_{D}(Z_{i1}) and μ_{D}(Z_{i2}). Simulation and empirical studies are provided as well.