Lecture notes:

  Intro to overlap issues and propensity scores
Lecture code
Code to match teffects command manually

More advanced:

Propensity score and mathing estimators
Lecture code


Brief overview (see PDF files for details  and code to replicate teffects command): Stata treatment effects are implemented with the teffects command, which is a great way of introducing semiparametric estimation of causal effects and issues of lack of overlap (common support) -- issues about regression adjustment in general, or the additional assumptions needed to obtain average treatment effects with linear/OLS models. The idea behind teffects is that causal effects are nonparametrically identified, so they can also be estimated nonparametrically. 

Stata’s teffects command estimates Average Treatment Effects (ATE), Average Treatment Effects on the Treated (ATET), and potential-outcome means (POMs). What all these mean exactly can be somewhat difficult to understand at first. The command uses several methods to obtain treatment effects: regression adjustment (not the standard version), inverse probability weighting (IPW), and a combination of both (“doubly robust” methods): inverse probability weighting plus regression adjustment (IPWRA) and augmented inverse-probability weighting (AIPW). Propensity score matching and other matching estimators are part of teffects as well (teffects psmatch, teffects nnmatch), including postestimation commands to check overlap and other useful statistics. These methods assume ignorability or conditional independence  or selection on observables, whichever is the term you prefer (and overlap for teffects ra). Note that Stata also has a eteffects —note  the “e” — command that doesn’t assume ignorability — that is, treatment assignment is correlated with treatment (the endogenous variable in econ jargon). It uses a control function approach. You can estimate (some) two-stage residual inclusion models (2SRI) with eteffects. 

It’s easier to understand the logic of teffects  if you replicate the multi-step estimation using standard regression step by step. Estimates will match but SEs will be  different.  teffects estimates all steps simultaneously using generalized method of moments estimation, GMM.  The discussion of causal effects in Wooldridge (2010), Chapter 21, is helpful to understand treatment effects implementation in Stata, although the implementation follows straight from the causal inference literature in statistics.