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[[File:Parallel Trend Assumption.png|right|thumb|320px| Illustration of the parallel trend assumption]]
All the assumptions of the [[Ordinary least squares#Assumptions|OLS model]] apply equally to DID. In addition, DID requires a '''parallel trend assumption'''. The parallel trend assumption says that <math>\lambda_2 - \lambda_1</math> are the same in both <math>s=1</math> and <math>s=2</math>. Given that the [[#Formal Definition|formal definition]] above accurately represents reality, this assumption automatically holds. However, a model with <math>\lambda_{st} ~:~ \lambda_{22} - \lambda_{21} \neq \lambda_{12} - \lambda_{11}</math> may well be more realistic. In order to increase the likelihood of the parallel trend assumption holding, a difference-in-differences approach is often combined with [[Matching (statistics)|matching]].<ref>{{cite journal |first1=Pallavi |last1=Basu |first2=Dylan |last2=Small |year=2020 |title=Constructing a More Closely Matched Control Group in a Difference-in-Differences Analysis: Its Effect on History Interacting with Group Bias |journal=[[Observational Studies]] |volume=6 |pages=103–130|doi=10.1353/obs.2020.0011 |s2cid=221702893 |url=https://obsstudies.org/wp-content/uploads/2020/09/basu_small_2020-1.pdf }}</ref> This involves 'Matching' known 'treatment' units with simulated counterfactual 'control' units: characteristically equivalent units which did not receive treatment. By defining the Outcome Variable as a temporal difference (change in observed outcome between pre- and posttreatment periods), and Matching multiple units in a large sample on the basis of similar pre-treatment histories, the resulting [[Average_treatment_effect|ATE]] (i.e. the ATT: Average Treatment Effect for the Treated) provides a robust difference-in-differences estimate of treatment effects. This serves two statistical purposes: firstly, conditional on pre-treatment covariates, the parallel trends assumption is likely to hold; and secondly, this approach reduces dependence on associated ignorability assumptions necessary for valid inference.
As illustrated to the right, the treatment effect is the difference between the observed value of ''y'' and what the value of ''y'' would have been with parallel trends, had there been no treatment. The Achilles' heel of DID is when something other than the treatment changes in one group but not the other at the same time as the treatment, implying a violation of the parallel trend assumption.
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A software example application of this research is found on the [[Stata]]'s command -diff- <ref name="StataDiff">{{cite journal |first1=Juan M.|last1=Villa |year=2016|title=diff: Simplifying the estimation of difference-in-differences treatment effects |journal=The Stata Journal |volume=16 |issue=1 |pages=
==See also==
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==Further reading==
*{{cite book |last1=Angrist |first1=J. D. |last2=Pischke |first2=J. S. |year=2008 |title=Mostly Harmless Econometrics: An Empiricist's Companion |publisher=Princeton University Press |isbn=978-0-691-12034-8 |pages=227–243 |url=https://books.google.com/books?id=ztXL21Xd8v8C&pg=PA227 }}
*{{cite book | first1 = Arthur C. |last1=Cameron |first2=Pravin K. |last2=Trivedi |year=2005 |title=Microeconometrics: Methods and Applications |publisher=Cambridge university press |isbn=9780521848053 |doi=10.1017/CBO9780511811241 |pages=768–772 |s2cid=120313863 |url=https://api.semanticscholar.org/CorpusID:120313863 }}
*{{cite journal |last1=Imbens |first1=Guido W. |first2=Jeffrey M. |last2=Wooldridge |year=2009 |title=Recent Developments in the Econometrics of Program Evaluation |journal=[[Journal of Economic Literature]] |volume=47 |issue=1 |pages=5–86 |doi=10.1257/jel.47.1.5 |url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:3043416 }}
*{{cite journal |first1=Jon |last1=Bakija |first2=Bradley |last2=Heim |title=How Does Charitable Giving Respond to Incentives and Income? Dynamic Panel Estimates Accounting for Predictable Changes in Taxation |journal=NBER Working Paper No. 14237 |date=August 2008 |doi=10.3386/w14237 |doi-access=free }}
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