Propensity Score Matching (PSM) Reference

2 Assumptions for treatment to be strongly ignorable (Rosenbaum Rubin 1983) thereby making PSM appropirate:
CIA (Conditional Independence Assump.): Conditional on observables (Xs) outcomes (Ys) are independent of treatment status (T=0 or 1).
 Common Support: For each value of observables (Xs) there is a positive probability of being treated and untreated. In other words observe all levels of X in both groups (treated and untreated).  

Pitfall--Curse of Dimensionality
The more observables the more difficult to find close matches across all Xs-hence need PS to gauge 'closeness'.

Matching Algorithms
Nearest Neighbor Matching
Radius Matching (calipers)
Kernel/local-linear matching (non-par that compares treated with w.avg. all all no treated; weighted by PS proximity.
Suggested test for robustness (Heinrich below)is to try all of them.(?)


Assumption and Spec Tests
For specification of the selection equations follow same rules as one would any other regression.
No great way of testing validity of CIA but can use to institutional knowledge to argue basis.
For common support one can test do an F-test (Hotelling test) on the joint X differences between treated and untreated.


Unobserved Heterogeneity and relaxing CIA
CIA can be somewhat relaxed if use DD on outcome comparisons.  Of course this requires the assumption that if there was selection on non-observables the non-observables be *time invariant*.  If that could strongly be argued then there is a strong case.  (requires panel data too).



PSM References: 
Heinrich, Maffioli, Vazquez, "A Primer for Applying Propensity-Score Matching", IDB Working paper 2010.
Angrist and Pirschke, "Mostly Harmless Econometrics", (Book) 2009.
Rosenbaum's "Observational Studies" (Book), 2002.