Is -xtregar- a large N, or a large T estimator?

[Joro Kolev]

Good evening,

I spent about 40 min reading the manual, and reading Baltagi and Wu (1999), and I was not able to determine with probability one whether "[XT] xtregar -- Fixed- and random-effects linear models with an AR(1) disturbance"
is an estimator which is consistent as the cross sectional dimension grows large, or is consistent as the time series dimension grows large.

I guessed it is the latter, it is consistent as the time series dimension grows large, but this was mostly because both Stata and Baltagi and Wu (1999) give as examples the Grunfeld data, and it is data that is with large time series dimension.

Does anybody know for sure?

Baltagi, B. H., and P. X. Wu. 1999. Unequally spaced panel data regressions with AR(1) disturbances. Econometric Theory 15: 814-823.

[Jeff Wooldridge]

Joro: Frankly, your question is the main reason I avoid this command. The underlying theory doesn't take a stand on how the asymptotics is being done -- and it's not a trivial point, as you clearly realize. I think it was written in the spirit of, "Here is a problem and we'll solve it with (feasible) GLS" without worrying about the asymptotics. I think there are assumptions where one could do large N, fixed T, or fixed T, large N. But I haven't done it.

For large N, small T, I use xtgee, where we can specify an AR(1) "working" correlation matrix. Very importantly, xtgee allows for vce(robust). Unfortunately, none of the Stata commands sureg, xtgls, and xtregar support robust variance matrices. They all should, because it's quite likely that the variance-covariance structure isn't correct.

[Joro Kolev]

Professor Wooldridge, thank you for confirming that the literature behind -xtregar- does not really say under what asymptotics this is supposed to work. Regarding the rest of the points you have made:

1. -xtgee- and -xtregar- purport to do different things. -xtgee- allows for AR process or exchangeable correlation (i.e., random effect). -xtregar- on the other hand purports to be able to allow for both exchangeable correlation (i.e., random effect) and on the top of it an AR process.

2. Whoever programmed -xtgee- was way ahead of the time (at least 20 years ahead of the pack) by programming the robust option. This comes because in turn Liang and Zeger figured out way ahead of the pack (mid 80ies) that assuming some "reasonable" correlation structure and following up by robust variance will not break the computer. I sort of witnessed the watershed in econometrics and in the Stata community. When I wrote the first draft of my Kolev (2012) paper which was about year 2008, Stata could not calculate robust variance post -xtreg- and there were lots of econometricians/statisticians both online and in person saying that "calculating robust variance post random or fixed effects does not make any sense". So in the first draft I had to quasi demean manually and use -regress- to be able to obtain robust variance. In the last draft circa 2012, Stata could already calculate robust variance, and econometricians/statisticians seemed to be coming to terms with the notion that "you can try to be efficient, but you do not need to assume that you have gotten it right."

3. Stata Corp still has not come to terms with the fact that you might want to have robust variance post sureg, xtgls, xtregar, and reg3. For sureg, xtgls and reg3 I know how to do this manually by using the _robust programmers command, but it is pain in the neck. I even wrote a note for Stata Journal on how to robustify post xtgls (attached to this message) but they were not impressed. And I do not blame them, it is indeed pain in the neck. Similarly when I needed, I robustified the variance post sureg by _robust, but it is a pain, somebody needs to sit down and to put this into a routine.

Kolev, G.I., 2012. Underperformance by female CEOs: A more powerful test. Economics Letters, 117(2), pp.436-440.
Liang, K.-Y., and S. L. Zeger. 1986. Longitudinal data analysis using generalized linear models. Biometrika 73:13–22.
Zeger, S. L., and K.-Y. Liang. 1986. Longitudinal data analysis for discrete and continuous outcomes. Biometrics 42:121–130.
Zeger, S. L., K.-Y. Liang, and P. S. Albert. 1988. Models for longitudinal data: A generalized estimating equation approach. Biometrics 44: 1049–1060.

Joro: Frankly, your question is the main reason I avoid this command. The underlying theory doesn't take a stand on how the asymptotics is being done -- and it's not a trivial point, as you clearly realize. I think it was written in the spirit of, "Here is a problem and we'll solve it with (feasible) GLS" without worrying about the asymptotics. I think there are assumptions where one could do large N, fixed T, or fixed T, large N. But I haven't done it.

For large N, small T, I use xtgee, where we can specify an AR(1) "working" correlation matrix. Very importantly, xtgee allows for vce(robust). Unfortunately, none of the Stata commands sureg, xtgls, and xtregar support robust variance matrices. They all should, because it's quite likely that the variance-covariance structure isn't correct.

[Jeff Wooldridge]

Joro:

Point well taken on xtregar allowing for an exchangeable structure with an AR(1). With large N, small T, I would just use xtgee, corr(uns) to allow for any correlation structure. This will pick up all such structures. We still use a robust option to allow general heteroskedasticity.

In the first edition of my MIT Press book (2002) I proposed always using a fully robust vce when using any large-N procedure, which included SUR and panel data. But as you noted, these ideas have been around in the GEE literature for longer. Hal White's (1984) Asymptotic Theory for Econometricians book also includes this recommendation, and formula.

I used to do exactly what you did for xtreg before Stata allowed the cluster option!

Best, Jeff