Difference in Differences

Difference in differences is a quasi-experimental method.

Description

The method can be expressed as:

where A, B, C, and D represent sample statistics; and where A and C are from treated samples. There is an assumption of parallel trends: absent treatment, AC should be parallel to BD as average change would have been equivalent.

The interpretation becomes:

• B is the baseline average
• D-B is the time trend
• A-B is the difference between groups
• (C-A)-(D-B) is the difference in differences

Implementation

The method is generally implemented using OLS regression.

Given indicators T where T_t=1 if time period t is the second one, and S where S_i=1 if observation i is in the treatment group, fit the following model:

Y_it = β₀ + β₁T_t + β₂S_i + β₃(T_tS_i) + ϵ_it

The coefficients can be interpreted as:

• β₀ is the baseline average

• β₁ is the time trend

• β₂ is the difference between groups

• β₃ is the difference in differences

Fixed effects for both units and time periods are generally included, leading to these being called two-way fixed effects models.

Alternatively, calculate the first differenced outcomes as Y'_i = Y_i2 - Y_i1. Now fit the model:

Y'_i = β₁ + β₃S_i + (ϵ_i2 - ϵ_i1)

Once again, β₃ is the difference in differences.

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