Propensity score in observational studies
The classic paper, “The central role of the propensity score in observational studies for casual effects“ by Rosenbaum & Rubin, is cited in many of the applied econometrics papers that deal with causality. In a randomized experiment, the outcome of the treated group and control group can be directly compared because the groups are likely to be similar. In an observational study, one does not have this luxury and in almost all the cases, the treated group and the control group vary in their composition. Hence the need for a technique to compute the treatment effect where there is a selection bias. This paper introduces “propensity score” as a solution to the selection bias.
The paper introduces the following terms :

Balancing score b(x) : It is a function of the observed covariates x such that conditional distribution of x given b(x) is same for treated (z=1) and control (z=0) units. The motivation behind using the term balancing' is that, for two set of units with the same balancing score, the treatment and control response can be directly compared.

Propensity score e(x) : It is the coarsest balancing score Pr(z=1x) and is defined as the propensity toward exposure to treatment 1 given the observed covariates x.

Strongly ignorable treatment assignment : If the assignment given a vector of covariates v is independent of the response
The paper presents five theorems whose conclusions are the following :

The propensity score is a balancing score. If a subclass of units or a matched treatmentcontrol pair is homogeneous in e(x), then the treated and control units in that subclass or matched pair will have the same distribution of x.

Any score that is finer than the propensity score is a balancing score. The propensity score is the coarsest. The practical importance of this theorem is that it is sometimes advantageous to subclassify or match not only for e(x) but for other functions of x as well, in particular, such a refined procedure may be used to obtain estimates of the average treatment effect in subpopulations defined by components of x

If treatment assignment is strongly ignorable given x, then it is strongly ignorable given any balancing score.This means that b(x) is sufficient to produce unbiased estimates of the average treatment effect

At any value of a balancing score, the difference between the treatment and control means is an unbiased estimate of the average treatment effect at that value of the balancing score if treatment assignment is strongly ignorable. Consequently, with strongly ignorable treatment assignment, pair matching on a balancing score, subclassification on a balancing score and covariance adjustment on a balancing score can all produce unbiased estimates of treatment effects.

Using sample estimates of balancing scores can produce sample balance on x
The paper concludes with a section that presents three applications of propensity scores to observational studies. The first application is the use of propensity scores to construct matched samples from treatment groups. The second application is related to subclassification of propensity scores and the third application is about covariance adjustment.