First Difference Models vs. Fixed Effects in Panel Data

[Jun Park]

Hello,

I am working with individual-level panel data to examine how political ideology varies with income and other financial variables. However, I am struggling to decide between using individual fixed effects or first differences (while including additional dummies like year or region).

One point of confusion is why some researchers use fixed effects while others include dummies instead. If fixed effects account for time-invariant individual characteristics, what is the advantage of using first differences with additional dummies? Under what circumstances would one approach be preferable to the other?

I would appreciate any insights on the trade-offs between these methods.

[Carlo Lazzaro]

Jun:
the -fe- estimator wipes out time-invariant variables
Thereforere, no coefficient for time-invariant predictor will be returned.
That said, I would consider -xtreg- first, assuming that you're dealing with a N>T panel dataset.

[Maxence Morlet]

Dear Jun,

Carlo gave you a great answer to your question. I might just quickly add that first difference also wipes out time-invariant variables. For T=2, both estimators will yield perfectly identical results.

It also sounds like you are referring to the equivalence between the fixed effects estimator, and the least squares dummy variable estimator. This equivalence is given by the Frisch Waugh Lovell theorem.

In first difference, the identification assumption is that the first difference in the error is uncorrelated with the (partialled out) first difference in the regressor.

[Jeff Wooldridge]

Just to add a bit more: Unless T is very large, you should include a full set of time dummies whether doing FE or FD. As Maxence pointed out, like FE, FD removes individual fixed effects. If one is use FD and adding things like regional dummies, then you are allowing trends to differ at the regional level. This is common in difference-in-differences settings to relax parallel trends, and can be useful more generally. You can achieve this in the FE setting by using, say, i.region#c.year, which estimates a separate linear trend for each region. You still want to included i.year to allow full flexibility in the average trend.

[Jun Park]

Thank you everyone! Is there a reason one should prefer the FD model over the FE model and vice versa?

Jeff: "Unless T is very large, you should include a full set of time dummies whether doing FE or FD": If T is very large, then are you talking about concerns over degrees of freedom?? I thought if T is very large, then one should definitely control for time trend. Is there a reason why some might not do that when T is very large?

[Jeff Wooldridge]

Jun: I'm thinking of cases where you have T so large that estimating an intercept for each different time period may be too much -- such as if you have daily or weekly data. Ideally, you should use time fixed effects. And you have to be clear about what you mean by "trend." The time FEs allow very flexible trends. Less desirable are functional forms such as linear, as these might not hold over the entire stretch of the data.