With stratification, observations are sorted from lowest to highest estimated propensity score. The comparison units with an estimated propensity score less than the minimum (or greater than the maximum) estimated propensity score for treated units are discarded. The strata, defined on the estimated propensity score, are chosen so that the covariates within each stratum are balanced across the treatment and comparison units (we know such strata exist from step one). Based on equation (2), within each stratum we take a difference in means of the outcome between the treatment and comparison groups, and weight these by the number of treated observations in each stratum. We also consider matching on the propensity score. Each treatment unit is matched with replacement to the comparison unit with the closest propensity score; the unmatched comparison units are discarded (see Dehejia and Wahba 1997 for more details; also Rubin 1979, and Heckman, Ichimura, and Todd 1997). ace payday loans
There are a number of reasons to prefer this two-step approach rather than estimating equation (1) directly. First, tackling equation (1) directly with a non-parametric regression would encounter the curse of dimensionality as a problem in many data sets, including ours, which have a large number of covariates. This is also true for estimating the propensity score using non-parametric techniques. Hence, we use a parametric model for the propensity score. This is preferable to applying a parametric model to equation (1) directly because, as we will see, the results are less sensitive to the logit specification than regression models, such as those in Table 2 (and because there is a simple criterion for determining which interactions to add to the specification).
Finally, depending on the estimator one adopts (e.g., stratification), an extremely precise estimate of the propensity score is not even needed, since the process of validating the propensity score produces at least one partition structure which balances pre-intervention covariates across the treatment and comparison groups within each stratum, which (by equation (1)) is all that is needed for an unbiased estimate of the treatment impact.