Stationary variables with Fixed Effects vs First Differences

[Kasper Rudy]

Hey, Gents!

Just a quick question.

I have some stationary variables in my dataset. Does this matter regarding the choice of a fixed effects estimation vs a first differences in a panel data analysis?

Hoping for an answer! Kind regards, Kasper

[Andrew Musau]

Most applications of FE/FD that I am aware of do not test for unit roots and cointegration of panel data. What you should be more concerned with is whether your regressors are time-varying because FD/FE in this context is aimed at eliminating the unobserved time-invariant individual specific effect.

As far as the choice of FD/FE goes, it matters little which you use. If you are manually estimating the FD model, you just have to be aware that with more than 2 repeated cross-sections (T>2), there are T-1 observations for each individual, and the transformed errors after taking first-differences will be autocorrelated. OLS will therefore be inefficient and the resulting standard errors biased. Therefore, you need to estimate with GLS (in this case, the estimator is feasible because the error covariance matrix does not depend on unknown parameters).

Back to the stationarity issue in panel data, you may be interested in the paper Spurious regression and residual-based tests for cointegration in panel data, Kao (1999), Journal of Econometrics.90 (p. 1-44).

http://faculty.maxwell.syr.edu/cdkao/working/je.pdf

Kao shows that even though non-stationarity in panels may result in spurious regressions, the FE estimator is consistent.

[Sebastian Kripfganz]

Actually, FD-GLS for a balanced panel model is identical to the FE estimator. You may find the following related Statalist discussion helpful:
Fixed effects and First Difference Estimators

[Andrew Musau]

Correct, FD-GLS for balanced panels is more easily calculated as the "within-groups" (FE) estimator to which it is identical. However, in practice, one may write a program that implements the former less-efficient algorithm.