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==General definition==

==General definition==

[[File:Illustration of Difference in Differences.png|thumb|upright=1.3|链接=Special:FilePath/Illustration_of_Difference_in_Differences.png]]

[[File:Illustration of Difference in Differences.png|thumb|upright=1.3|链接=Special:FilePath/Illustration_of_Difference_in_Differences.png]]

Difference in differences requires data measured from a treatment group and a control group at two or more different time periods, specifically at least one time period before "treatment" and at least one time period after "treatment." In the example pictured, the outcome in the treatment group is represented by the line P and the outcome in the control group is represented by the line S. The outcome (dependent) variable in both groups is measured at time 1, before either group has received the treatment (i.e., the independent or explanatory variable), represented by the points ''P''₁ and ''S''₁. The treatment group then receives or experiences the treatment and both groups are again measured at time 2. Not all of the difference between the treatment and control groups at time 2 (that is, the difference between ''P''₂ and ''S''₂) can be explained as being an effect of the treatment, because the treatment group and control group did not start out at the same point at time 1. DID therefore calculates the "normal" difference in the outcome variable between the two groups (the difference that would still exist if neither group experienced the treatment), represented by the dotted line ''Q''. (Notice that the slope from ''P''₁ to ''Q'' is the same as the slope from ''S''₁ to ''S''₂.) The treatment effect is the difference between the observed outcome (P₂) and the "normal" outcome (the difference between P₂ and Q).

Difference in differences requires data measured from a treatment group and a control group at two or more different time periods, specifically at least one time period before "treatment" and at least one time period after "treatment." In the example pictured, the outcome in the treatment group is represented by the line P and the outcome in the control group is represented by the line S. The outcome (dependent) variable in both groups is measured at time 1, before either group has received the treatment (i.e., the independent or explanatory variable), represented by the points ''P''₁ and ''S''₁. The treatment group then receives or experiences the treatment and both groups are again measured at time 2. Not all of the difference between the treatment and control groups at time 2 (that is, the difference between ''P''₂ and ''S''₂) can be explained as being an effect of the treatment, because the treatment group and control group did not start out at the same point at time 1. DID therefore calculates the "normal" difference in the outcome variable between the two groups (the difference that would still exist if neither group experienced the treatment), represented by the dotted line ''Q''. (Notice that the slope from ''P''₁ to ''Q'' is the same as the slope from ''S''₁ to ''S''₂.) The treatment effect is the difference between the observed outcome (P₂) and the "normal" outcome (the difference between P₂ and Q).

right|thumb|320px| Illustration of the parallel trend assumption

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-difference approach is often combined with [[Matching (statistics)|matching]].<ref name=":2">{{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|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-difference 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.   

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-difference approach is often combined with [[Matching (statistics)|matching]].<ref name=":2">{{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|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-difference 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.   

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! !!新泽西州

! !!新泽西州

|宾夕法尼亚州

|'''宾夕法尼亚州'''

|差异

|'''差异'''

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| February || 20.44 || 23.33 || −2.89

| February || 20.44 || 23.33 || −2.89