@TechReport{iza:izadps:dp4304,
            	author={Heckman, James J. and Todd, Petra E.},
            	title={A Note on Adapting Propensity Score Matching and Selection Models to Choice Based Samples},
            	year={2009},
	month={Jul},
            	institution={Institute of Labor Economics (IZA)},
            	address={Bonn},
            	type={IZA Discussion Paper},
            	number={4304},
            	url={https://www.iza.org/index.php/publications/dp4304},
	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.},
	keywords={propensity scores;matching models;choice-based sampling;selection models},
            }