Estimating Heterogeneous Treatment Effects with Observational Data

Individuals differ not only in their background characteristics but also in how they respond to a particular treatment, intervention, or stimulation. In particular, treatment effects may vary systematically by the propensity for treatment. In this paper, we discuss a practical approach to studying heterogeneous treatment effects as a function of the treatment propensity, under the same assumption commonly underlying regression analysis: ignorability. We describe one parametric method and two nonparametric methods for estimating interactions between treatment and the propensity for treatment. For the first method, we begin by estimating propensity scores for the probability of treatment given a set of observed covariates for each unit and construct balanced propensity score strata; we then estimate propensity score stratum-specific average treatment effects and evaluate a trend across them. For the second method, we match control units to treated units based on the propensity score and transform the data into treatment-control comparisons at the most elementary level at which such comparisons can be constructed; we then estimate treatment effects as a function of the propensity score by fitting a nonparametric model as a smoothing device. For the third method, we first estimate nonparametric regressions of the outcome variable as a function of the propensity score separately for treated units and for control units and then take the difference between the two nonparametric regressions. We illustrate the application of these methods with an empirical example of the effects of college attendance on women's fertility.