Time invariant variable estimation in panel data analysis, what estimator to use?

[Henk Hollander]

Hi everyone.
I have an interesting panel data set but I am afraid one of the main interesting independent variables is time-invariant, but I would still like to discuss it in my thesis. This would disappear using fixed effects or first differences. How to proceed?

Formula (i = company, t = time):
Y_it = b0 + b1 * X1_it + b2 * X2_i + b3 * X2_i * X1_it + u_it

Objective: I am interested in mainly b3

So X2 would be if a company (i) is in the USA (dummy X2) or not, and due to data set limitations I probably expect the variable to be time invariant in my dataset. I wish to compare it to the EU.

t is more than 2 years (so no diff and diff?)

I could restrict _i to companies of a certain country, but then I can only get a feel for if they are different and not if they are statically significantly different right?

I could get time invariant variables using RE, but I am quite sure that it wouldn't pass the houseman test.

Yours sincerely,
A student who needs help for his thesis

p.s. for others this link might help: https://www.stata.com/manuals/xtxtreg.pdf . I already had a look at it.

[Clyde Schechter]

I don't see the problem. Is X1 also time-invariant? If, as I hope and suspect, it is not, then the interaction X2*X1 will vary within firms over time, so b3, the object of your investigation, will be estimable with a fixed-effects model.

If X1 is also time-invariant, then nothing in your model varies over time and a fixed effects approach has nothing to offer.

[Henk Hollander]

Thank you for the speedy response.
X1 wont be time invariant.
Would it be a problem if X2 is a dummy?
Does the interaction term still vary within firms if X2 does not vary within firms.

[Clyde Schechter]

It makes no difference whether X2 is a "dummy." In general, if X1 varies within firms, but X2 does not, then X1*X2 will vary within firms. I say "in general" because one can construct special examples where that fails, but they are artificial and situations like that seldom arise in real life. For example, if X2 is a dichotomous 0/1 variable, and if X1 varies within firms, but only within those firms where X2 = 0, and X1 is time-invariant whenever X1 = 1, then X1*X2 will also be time-invariant. But unless you have some artificial situation like that, X1*X2 will vary over time within firms.

[Henk Hollander]

mr Schechter, I thank you for you time and wisdom.

[Jeff Wooldridge]

If you want to see a coefficient on X2, and still obtain the FE estimates on everything else (time-varying, including X1*X2), you can use the correlated random effects estimator. You can create the average over time for each individual using egen, and then add these time averages to the equation and estimate by random effects or even pooled OLS. It's a bit tricky with an unbalanced panel, but still not too difficult. Plus, with the vce(cluster id) option you can get a regression-based, robust Hausman test.

[Sebastian Kripfganz]

If you want to see a coefficient on X2, and still obtain the FE estimates on everything else (time-varying, including X1*X2), you can use the correlated random effects estimator. You can create the average over time for each individual using egen, and then add these time averages to the equation and estimate by random effects or even pooled OLS. It's a bit tricky with an unbalanced panel, but still not too difficult. Plus, with the vce(cluster id) option you can get a regression-based, robust Hausman test.

With Stata 18.5 (StataNow), you can do this now without the manual creation of the averages by using xtreg, cre.