It is an econometric method used to evaluate the causal impact of a specific intervention or treatment. This approach compares the changes in outcomes of a group receiving the treatment with the changes in a control group that does not receive it, before and after the implementation of the intervention.
The main advantage of DiD is its ability to control for unobservable factors that might affect both groups similarly over time, allowing for the isolation of the true treatment effect.
Equations and Assumptions
Parallel Trends Assumption: The time trends of the treatment and control groups would have been parallel in the absence of the treatment.
Data Generating Model:
\[ Y_{it} = \alpha + \beta T_i + \gamma Post_t + \delta (T_i \times Post_t) + \epsilon_{it} \]
Where:
- \( Y_{it} \) is the outcome for individual \( i \) at time \( t \).
- \( T_i \) is an indicator variable that equals 1 if the individual belongs to the treatment group and 0 if they belong to the control group.
- \( Post_t \) is an indicator variable that equals 1 if time \( t \) is after the treatment and 0 if it is before.
- \( \delta \) is the parameter of interest that measures the treatment effect.
- \( \epsilon_{it} \) is the error term.
Difference-in-Differences (DiD) Method:
\[ DiD = (\overline{Y}_{Treated, After} - \overline{Y}_{Treated, Before}) - (\overline{Y}_{Control, After} - \overline{Y}_{Control, Before}) \]
Interpretation of DiD with Initial Difference:
Even if there is an initial difference between the groups (\( \overline{Y}_{Treated, Before} \neq \overline{Y}_{Control, Before} \)), the DiD estimator adjusts for this difference by subtracting the changes in both groups, thus isolating the treatment effect.