Fixed Effects Panel Regression with large N and small T: Cross Dependence and time fixed effects

[Nikolaus Schueler]

Dear STATA community,

I am new to this forum and I hope you can help me with my problem.

i am investigating expenditures of municipalities in a german state between 2009 and 2020 (N = 400, T = 11 years) by using a fixed effect panel data regression. One central point of interest is the investigation of a policy measure that took place in 2015 and lasted until 2020. I am planning on modeling this measure with a explanatory dummy variable which turns to 1 in 2015. This is the reason why I am using a fixed effects model as I am intrested in intra-municipal changes over time especially due to this external shock (hausmann test recommended it as well).

So far, I've been using the xtreg, fe command with cluster robust standard errors (vce(cluster id)).

After the regression the testparm i.year command is telling me that there are time fixed effects which need to be included in the model. Furthermore, the Pesaran test (xtcsd, pesaran abs) identifies cross level dependency which - as I have read - is quite ususal when working with administrative observation units.

I was thinking about using Driscoll and Kraay (1998) robust standard errors (xtscc) in orer to controll for cross level dependency, but according to the literature these only apply for cases with large T and small N.

Both the cross dependence issue and the problem with time fixed effects might be solved by including year dummies, but ,first, I don't know how to interpret the dummies (or whether they are interpretable at all) and, second, I think there will be problems interpreting the policy dummy as the effect might be (partially) absorbed by the year dummy.

My questions in particular:

• Are there standard errors which controll for cross level dependencies in cases with large N and small T?
• Do you know other ways of controlling for time fixed effects when using a fixed effects model?

I would be really thankful If you could help me. If you need any further information please tell me!!

Best regards
Niko

[Maxence Morlet]

Do you only have 2 sources of variation? (i.e. municipality and time)

If there is variation within municipality of your treatment, a treatment dummy specified as

treated*post

should not drop due to collinearity.

You may want to try the community contributed command

reghdfe

instead of xtreg, fe.

But in practice, in your case especially, the way to go is indeed two way fixed effects with cluster robust standard errors. Interpreting coefficient on year dummies is rarely of interest, but here it is: Stata will omit one dummy, that will be the base year. All coefficients on other years will be the change, relative to the base year, in the dependent variable (on average).

[Nikolaus Schueler]

Dear Maxence,

thank you for your quick response - especially regarding the interpretation of year dummies!

Do you only have 2 sources of variation? (i.e. municipality and time)

There are six explanatory variables on municipal level plus policy dummy plus time. In 2015 the policy dummy turns to 1 for ALL municipalities.

If there is variation within municipality of your treatment, a treatment dummy specified as Code:treated*post should not drop due to collinearity.

I am not sure if I understand this correctly. So instead of using a 0/1-dummy, do you recommend modelling the treatment on the ID-level in a different way?

And regarding the implementation of time and cross fixed effects: Do you think a solution might be to use a lagged dependent variable as additional explanatory variable?

Best regards
Niko

[Carlo Lazzaro]

Nikolaus:
1) Maxence was wisely suggesting the so-called two-way fixed effect regression (municipalities and time), whwreas, if I get you right, the 6 variables are the regressors to be plugged in the right hand-side of your regression equation.
2) The policy-related categorical variable should be set to 1 from 2015 to 2020.
3) The -fe- estimator wipes out time-invariant variables and works at its best when there is within-panel variation as far as the time-varying variables are concerned.
4) The usual way to test -i.timevar- is via -testparm-. In addition, -i.timevar- should be added by default among the predictors of -xtreg.,fe-.
5) I do agree with Maxence's recommendation about going cluster-robust standard errors.
6) Going -xtabond- would mean to eneter the really tricky realm of dynamic panel data regression. In addition, -xtabond- requires no correlation of the epsilon error (that you seem to have).

[Nikolaus Schueler]

Dear Carlo,

thank you very much for your hints! And yes, those six variables are explanatory variables on the right hand side.

So basically, you recommend using a two way fixed effects model (xtreg, fe) by implementing year dummies (i.year) as predictors with cluster robust standard errors (vce(cluster id)). You don't recommend using Driscoll and Kraay (1998) robust standard errors (xtscc) because there is no large T in relation to the size of N, right?

And one more question: Do you know of any other way of modelling such kind of policy measure?

Again thank you for your effort!

Best regards
Niko

[Carlo Lazzaro]

Nikolaus:
1) your intrepretation is correct: -xtreg,fe- with -i.year- among predictors and cluster-robust standard errrors. No -xtscc-;
2) Perhaps such kind of policy measure can be invedtugated with a difference-in-difference approach, whose set up requires two groups (treated/not treated) and a time variabale (before/after the measure). Unhfortunately, I cannot say if this requirements are fulfilled by your data.

[Nikolaus Schueler]

Thanks Carlo, but I don't think DiffInDiff works for my research question because the treatment (policy dummy 0/1) applies for all IDs - so there is no differentiation between groups, but only between preiods of time.