TWFE for pooled cross sections

[Taiba Chau]

Hi, I currently have a dataset looking at 500 days, which is from 3rd April 2018 to 16th August 2019. That is for two seasons, spring and summer. This is recorded for 30000 homes in 10 EU countries. This is based on whether the homes are exposed to a specific program or not. If I were to write a standard TWFE estimator just based on this would it be:

Here c denotes the home, t denotes the day. Where ac denotes the home dummy.
Then Wct is dummy for whether they undertook the program. This is the interaction term for the time dummy and treatment dummy.
pct are the control and u is the error term.

I am wondering if this is correct?

[Clyde Schechter]

There is no u in the equation you show. The error term in that model is epsilon_ct.

[Jared Greathouse]

Note that I'm not an econometrician (yet), and I know lots of this can sometimes be subjective.... but, if this were me writing this, I'd write it what I think is clearer notation.

I'd likely write something like

Y_{it}= Beta_1(I)+X_{it} +delta+epsilon_{it}


Where in order we have our outcomes, our treatment, a matrix of covariates, delta which reps unit and time dummies, and our error term.

I'm on my phone so forgive me abusing notation slightly, but to me, the notation should be written as clearly as possible. I know the appeal of using lots of sexy Greek letters and notation (you should see the papers I write!!), but whatever notation we use it should be as clear and well defined as possible. I think the multiplication of pho and gamma(?), for example, is a little confusing. It implies an interaction- and if you're doing that, it doesn't immediately seem to make sense, and if not, it seems like you are.

Now perhaps you explicate these terms in your paper, so I could be misunderstanding, but whatever notation you do use, just make sure it's clear and well defined.

I'm certainty open to correction though, from someone who's better at notation than I am.

[Taiba Chau]

Sorry yes epsilon_ct is the error term. I was just wondering if this is correct

[Maxence Morlet]

Wooldridge (2021): (PDF) Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimators (researchgate.net)

Absolutely amazing, clear and concise paper on TWFE and equivalences. I'd definitely recommend that you give it a read Taiba Chau

[Taiba Chau]

I tried running the regression for it in stata. But I have tried running a few regressions:

reg outcome o030418.date#o1.treat i.date treat if date>=030418 & date<=160819, robust
This is the regression looking at the whole time period which is a balanced panel

I then ran two further regressions:
reg outcome o060518.date#o1.treat i.date treat if date>=060518 & date<=300918, robust
This regression is looking at the intervention period.

reg outcome o121118.date#o1.treat i.date treat if date>=121118 & date<=030319, robust
This is another regression looking at a second intervention period.

Any ideas if these are okay and any ways to improve the analysis? Could I run other regressions that may be interesting?

I have seen a lot using xtreg but how would this relate to the interaction term?