A few more things about panel data estimation techniques.
Fixed-effects versus a dummy variable for each observation: These
techniques are equivalent. They give exactly the same estimates of coefficients, and
exactly the same standard errors, and exactly the same results when testing
hypotheses. The dummy-variable technique is computationally intensive, but it
automatically gives you estimates of the c i values. This might be useful if you’re
interested in knowing the distribution of c i . It’s also useful when you want to
account for two (or more) different sources of fixed effects. (Interestingly, while
these techniques are exactly equivalent, they yield different R 2 values—another
example of why you can’t interpret an R 2 coefficient on its own.)
Fixed-effects versus first-differences: Both estimation techniques offer
unbiased results whenever cov(xit ,c i ) ≠ 0 , as long as sequential exogeneity holds. In
many cases, there’s little reason to prefer one over the other. Some considerations:
1. We haven’t discussed any trends in the uit unobservables. Sometimes, luck in
one year might correlate with luck in other years: if this year you have a job
that pays unusually well, then next year you’ve got a good chance of being in
the same job with high wages. Sometimes not: if this year you have high
income because you won the lottery, then next year you go back to being
average. This issue of “serial correlation” is complex. Both FE and FD rely on
some assumptions, and I’m going to avoid going into detail about those. Firstdifferences might be more appropriate when the randomness tends to be
correlated over time, while fixed-effects might be more appropriate when
randomness tends to dissipate completely between periods.
2. When “strict exogeneity” fails, both FE and FD are biased. This serves as a
useful check of this assumption. Estimate the model both ways, and compare
the estimated coefficients. If strict exogeneity is satisfied, then you expect to
get similar results. If there is a violation, then both estimators are biased, but
in different ways. This will cause FE and FD to look very different.
3. When strict exogeneity fails, it’s usually easier to remedy the problem using
FD than FE.
Constant terms: If the model is yit = β1 + β 2 xit + … , then you’re not going to
recover β1 by doing regression with the fixed-effects or first-differences transformed
data. They cancel out of the transformations. If you capture a constant term, it
reflects the time trend.
When strict exogeneity fails: All of the techniques result in biased estimates, and
we have to resort to instrumental variables to correct the problem. We can combine
the IV technique with panel data techniques. However, we still need an instrument
that’s uncorrelated with the error term—or with the transformed error term, when
we’re doing panel data transformations. This is another reason why we might favor
one estimation technique. Fixed-effects requires that the instrument zit is
uncorrelated with u it = uit − ui = uit − T1 ∑ uis . Usually, this requires that zit is
uncorrelated with every single uis : past, present, or future.
First-differences, with instrumental variables, is unbiased if the instrument is zit is
unrelated to ∆ uit = uit − uit −1 . This is a small difference, but sometimes it comes in
handy: it requires that the instrument zit is uncorrelated with just uit and uit −1 .
In some cases, that means that we can use xit − 2 as our instrumental for Δxit . It’s
likely that the past xit − 2 is correlated with xit −1 and xit (and thus, with Δxit ) but not
with future uit or uit −1 , so not with Δuit . This situation is sometimes described as
“sequential exogeneity”.