DiD with multiple time periods

[Alexa Smith]

Hi there!

I am analyzing the effects of treatment on my outcome variable of interest. To do so I observe units from across different countries (cohorts) that became subject to treatment in different years (staggered treatment). To condition on parallel trends, I include unit-specific and country-specific covariates that might influence my outcome variable other than treatment itself. Now, I am pretty new to stata. However I know that csdid or xthdidregress both implement Callaway & Sant'Annas (2021) framework. Their framework heavily relies on within-cohort variation.

I realized that although my country-specific (macroeconomic) variables differ across countries, they remain identical for all units within the same cohort during each observation year, leading to no within-country variation. What should I do now? The macroeconomic variables are very important in explaining variation in the outcome variable, as the outcome is very sensitive towards them.

I read online that when including fixed effects in the model, this would ultimately cancel out the macroeconomic variables. But how do I know whether the command csdid or xthdidregress uses fixed effects? What if I persist on inducing macroeconomic variables? How can I control for this and the issue that entire cohorts might be fully omitted when computing the ATET's?

Tank you so much in advance!
I really appreciate the help.

[Jeff Wooldridge]

Callaway-Sant'Anna works by effectively using changes in the outcome from before to after the intervention. Because of this, time fixed effects are accounted for. It's similar to regression in levels where you included unit fixed effects, time fixed effects, and flexible treatment effects. Therefore, any aggregate variables assumed to affect the units in the same way get absorbed by the time fixed effects. Putting in time FEs is more general than controlling for macro variables that only change across t.

[Alexa Smith]

Thank you so much for your answer! Did I get it right that when employing the csdid command, FE are already incorporated in the model, which is why I do not need to control for cohort-specific differences and can simply leave them out (but keep the unit-specific variables)?

[Jeff Wooldridge]

Alexa, that's correct. I should've stated things more clearly. The differencing used by CS effectively remove those unobserved differences that change over time. Then, because CS essentially does regression analysis with these differences, the estimated intercepts from the long differencing are essentially the estimates on the time fixed effects. In the simplest case with T = 2, the regression-based version of CS would be identical to the first-differencing estimator with a time fixed effect (for the second period) and full individual FEs.

[Alexa Smith]

Thank you for your help! I really appreciate it!

[Alexa Smith]

The differencing used by CS effectively remove those unobserved differences that change over time.

I actually happen to have a follow-up question: Does the same argumentation apply in case I observe two distinct countries with differences in macroeconomic variables being subject to treatment at the same time, and thus, represent one cohort?