I recently have stumbled across a number of studies incorporating both difference-in-differences (DD) and propensity score methods. As discussed before, DD is a special case of fixed effects panel methods.
In the World Bank's publication "Impact Evaluation in Practice" they give a nice summary of the power of DD in identification of causal effects:
"...we can conclude that many unobserved characteristics of individuals are also more or less constant over time. Consider, for example, a person's intelligence or such personality traits as motivation, optimism, self-discipline, or family health history...Interestingly, we are canceling out(or controling for) not only the effect of observed time invariant characteristics but also the effect of unobserved time invariant characteristics such as those mentioned above"
So with DD we can actually control for unobserved characteristics that we may not have data on or maybe couldn't measure appropriately or even quantify! That's powerful. In this framework we are controlling for unobservable characteristics that may be contributing to selection bias, we are achieving identification of treatment effects in a selection on unobservables context.
On the other hand, with propensity score matching, we are appealing to the conditional independence assumption, the idea that matched comparisons imply balance on observed
covariates, which ‘recreates’ a situation similar to a randomized experiment where all subjects are essentially the same
except for the treatment(Thoemmes and Kim, 2011). Propensity score matching can identify treatment effects in a selection on observables context.
But, what if we combine both approaches. The Impact Evaluation book has a section on mixed methods that gives a really good treatment of the power of using both PSM and DD:
"Matched difference-in-differences is one example of combining methods. As discussed previusly, simple propensity score matching cannot account for unobserved characteristics that might explain why a group chooses to enroll in a program and that might also affect outcomes. By contrast, matching combined with difference-in-differences at least takes care of any unobserved characteristics that are constant across time between the two groups"
Below are several papers that utilize the combination of DD and PSM:
Does Matching Overcome Lalonde’s Critique of Nonexperimental Estimators? Jeffrey Smith and Petra Todd. University of Maryland. 2003
Do Agricultural Land Preservation Programs Reduce Farmland Loss? Evidence from a Propensity Score Matching Estimator
Xiangping Liu and Lori Lynch January 2010
Measuring the Impact of Meat Packing and Processing Facilities in the Nonmetropolitan Midwest: A Difference- in-Differences Approach
Georgeanne M Artz, Peter Orazem, Daniel Otto
November 2005 Working Paper # 03003
Iowa State University
How Effective is Health Coaching in Reducing Health Services Expenditures?
Yvonne Jonk, PhD,* Karen Lawson, MD,w Heidi O’Connor, MS,z Kirsten S. Riise, PhD,y David Eisenberg, MD,8z Bryan Dowd, PhD,z and Mary J. Kreitzer, PhD, RN, FAANw
Medical Care Volume 53, Number 2, February 2015
Impact Evaluation in Practice
Paul J. Gertler Sebastian Martinez, Patrick Premand,
Laura B. Rawlings and Christel M. J. Vermeersch
Default Book Series.December 2010