Estimation is in two steps. First, we estimate the propensity score for the sample of experimental treatment and non-experimental comparison units. We use the logistic model, but other standard models yield similar results. An issue is what functional form of the pre-intervention variables to include in the logit. We rely on the following proposition:
Proposition 2 asserts that, conditional on the propensity score, the covariates are independent of assignment to treatment, so that, for observations with the same propensity score, the distribution of covariates should be the same across the treatment and comparison groups. Conditioning on the propensity score, each individual has the same probability of assignment to treatment, as in a randomized experiment.
We use this proposition to assess estimates of the propensity score. For any given specification (we start by introducing the covariates linearly), we group observations into strata defined on the estimated propensity score and check whether we succeed in balancing the covariates within each stratum. We use tests for the statistical significance of differences in the distribution of covariates, focusing on first and second moments. If there are no significant differences between the two groups, then we accept the specification. If there are significant differences, we add higher-order terms and interactions of the covariates until this condition is satisfied. Section 5 shows that the results are not sensitive to the selection of higher order and interaction variables.
In the second step, given the estimated propensity score, we need to estimate a univariate non-parametric regression E(<Yi| T = j, p(Xt)), for j=0,1. We focus on simple methods for obtaining a flexible functional form, stratification and matching, but in principle one could use any one of the standard array of non-parametric techniques (e.g., see Hardle 1990). payday loan direct lenders