@techreport{NBERw15179,
 title = "A Note on Adapting Propensity Score Matching and Selection Models to Choice Based Samples",
 author = "James J. Heckman and Petra E. Todd",
 institution = "National Bureau of Economic Research",
 type = "Working Paper",
 series = "Working Paper Series",
 number = "15179",
 year = "2009",
 month = "July",
 URL = "http://www.nber.org/papers/w15179",
 abstract = {The probability of selection into treatment plays an important role in matching and selection models. However, this probability can often not be consistently estimated, because of choice-based sampling designs with unknown sampling weights. This note establishes that the selection and matching procedures can be implemented using propensity scores fit on choice-based samples with misspecified weights, because the odds ratio of the propensity score fit on the choice-based sample is monotonically related to the odds ratio of the true propensity scores.},
}