two way fixed effects: Mundlak specification (time effects and fixed effects)

[Anna Volkova]

Dear all,
I am working with panel data and would like to proceed with a two-way fixed effects model:
𝑌_it=𝛼_i+𝛾_t+𝛽𝑋_i+𝛿Z_𝑖𝑡+𝜖_𝑖𝑡
In my model, I have unobserved heterogeneity (a_i) and time effects (𝛾_t). Instead of using the within transformation, I want to apply the Mundlak specification to ensure the correlation between 𝛼_i and regressors (I do not want to use the within transformation as it will remove 𝑋𝑖 from my model). The Mundlak approach is to time-averages of Z_𝑖𝑡 as an additional regressor. Yet, I want to account for the unobserved time effects (𝛾_𝑡). Does it make sense to simply include time dummies (given that I have very few time periods, so I think it is rather efficient) in Mundlak's specification? So the final model I want to estimate is:
𝑌_it=𝛼_i+𝛾_tD_t+𝛽𝑋_i+𝛿Z_𝑖t+b\bar{Z_𝑖}+𝜖_𝑖𝑡 where 𝐷𝑡is the time dummy.
Would be thankful for any suggestions.

[Sebastian Kripfganz]

You can definitely include time dummies in your Mundlak specification, yes.

[Anna Volkova]

Sebastian Kripfganz thank you. If I do so, do I also have to include the time averages of time dummies (in an unbalanced case) in the model?

[Sebastian Kripfganz]

For the estimate of 𝛿 to be identical to the fixed-effects estimate, you also need to include the time averages of the time dummies (in unbalanced panels only), yes.

[Anna Volkova]

@Sebastian Kripfganz thanks again. If I may ask one more question for my research. I'm working with flight data and I want to see/check for which levels I should include the fixed effects. I have data regarding flight delays per multiple airlines and each airline obviously includes multiple aircraft. Thus, I was thinking about whether I should include fixed effects per the airline or per aircraft or maybe per itinerary (sequence of flights performed by aircraft within one day). I have noticed that adding fixed effects per airline and per aircraft leads to similar results but the estimates when I include fixed effects per itineraries are much larger. Are there any specification tests to compare the models?

[Sebastian Kripfganz]

I would not expect any aircraft-fixed effects beyond the airline-fixed effects, which is in line with your observation that the results are quite similar. Itinerary-fixed effects could make sense given that a tight itinerary can be a cause for delays. Since a given aircraft (typically) is just used by the same airline, a model with such itinerary-fixed effects could be seen as a more general model than one with only airline-fixed effects. Significant differences would then indicate that indeed these itinerary-fixed effects are important beyond the the airline-fixed effects. In principle, the Hausman approach could be used for comparing the two models since the estimator with itinerary-fixed effects would remain consistent but inefficient if there are only airline-fixed effects, while the estimator with airline-fixed effects would be efficient if there are indeed no itinerary-fixed effects but would become inconsistent otherwise.

[Anna Volkova]

Sebastian Kripfganz thank you! While implementing the Hausman test (to compare the estimates when using airline vs itinerary fixed effects), I got one more question. In my research, I want to use Mundlak specification, while the Hausman test in STATA is normally aimed at comparing fixed effects vs random-effects models. In the Mundlak setting, the procedure of Hausman is somewhat different: as far as I understand, if I want to perform the Hausman test in the Mundlak setting, I need to jointly test the coefficients of time-averaged regressors. How can I then adopt this to test itinerary vs aircraft fixed effects?

Is it maybe possible to run standard fixed effects with aircraft fixed effects:

xtreg, fe
estimates store fe1

then run standard fixed effects with itinerary fixed effects:

xtreg, fe
estimates store fe2

and use

hausman fe1 fe2

So in this way, I suppose I can choose between fe1 and f2 (even though time-invariant regressors drop out), thus I know which fixed effects are appropriate in my case. Having done this, I can proceed with the Mundlak estimation (to keep time-invariant variables in my model). Would this be a valid procedure?

Or should I simply estimate the two Mundlak models (so one with adding time averages for aircraft, another with adding time averages per itinerary) and then simply run?

hausman estimates_aircraft estimates_itinerary

[Sebastian Kripfganz]

I would do the last thing, and ideally use the suest command (with vce(robust) or vce(cluster)) instead of the hausman command to do the test. This would give you a "robust Hausman test".