Standard Error/Confidence Intervals for a Difference-in-differences analysis

I'm attempting a 'difference-in-differences' analysis of a health policy intervention.

Scenario: Health clinics are paid for the percentage of eligible patients who they give the right treatment to. The clinics are measured/paid separately for different treatment indicators (e.g. treatment for diabetes, for high blood pressure).

One year, they stopped payments for some of the indicators (but others continued). I want to measure if stopping the payment had an impact on performance (relative to the control group).

I have performance data for every single health clinic in the country - for the intervention and control group (both pre- and after- intervention.

My questions are:

1. If I have data for every single health clinic in the country (it's not a sample), should I still calculate the Standard Error and confidence intervals for the difference-in-differences (DiD) estimator?

2. How best should I calculate the SE and CI for the DiD estimator? Should I use just the mean average, or can I factor in the individual pairs of measurements (for each clinic)? There is a pre- and post- measurement, for the control and the intervention, for every single clinic.

3. Each clinic has varying numbers of eligible patients (I have the numerator/denominator for each); can I factor that into the estimation of the impact?

I'm doing the analysis in STATA.

2. The easiest way to implement DiD is to formulate it as a regression and just run OLS: $$Performance_{c,t} = \alpha + \beta_{1}*D_{after,c,t} + \beta_{2}*D_{intervention,c,t} + \beta_{3}*D_{after,c,t}*D_{intervention,c,t} +\epsilon_{c,t}$$
where $$D_{after,c,t}$$ is a dummy variable that is set to 1 for observations after the change and 0 before it and $$D_{intervention,c,t}$$ is a dummy variable that is set to 1 for observations in the intervention group.
$$\beta_{3}$$ is your DiD estimator and the regression will give you it's standard error, confidence interval, and t-statistic.
• Yes, ((InterventionAfter-ControlAfter)-(InterventionBefore-ControlBefore)) is one way to calculate the DiD estimator, but you should get the same number if you run the linear regression I specified with the standard OLS estimator of $(X'X)^{-1}X'y$. Any statistical software will easily give you the coefficient estimates and their t-stats and confidence intervals, there is no need to calculate them manually. I'm not going to write the formulas here, they get messy for multivariate regression. – Matt P Apr 10 at 15:20