# Fixed effects vs first difference

Or more generally, what are the reasons for the absolute dominance of the fixed effects estimators to control for unobserved heterogeneity in large N, short T panel settings?

I get that random effects imposes normality conditions which is awkward, but for instance if you compare FE with first difference, the latter is less efficient but only relies on the weak (rather than strong) exogeneity assumption. A priory, I would rather have a less efficient estimator with weaker identification assumption. Am I missing something?

I do not think the premise is correct. Following Brüderland and Volker in Best & Wolf The SAGE Handbook of Regression Analysis and Causal Inference [square brackets have my remarks]:

Both estimators require strict exogeneity [Fixed Effects (FE) and First Differences (FD)]. However, while FE builds on the assumption of no serial correlation prior to demeaning (see condition (15.10)), FD relies on no serial correlation in the differenced errors. The latter assumption is equivalent to very strong correlation in the untransformed errors. FE and FD therefore rely on assumptions that are opposite extremes.

So both estimators require strong exogeneity.

This being said FE has and serious advantages over FD. Again following Brüderland and Volker:

Any of the four basic within estimators is appropriate for dealing with time-constant confounders. Nevertheless, FE regression has some important practical advantages over the others. ...

FD might be preferable in the presence of strong serial correlation. However, besides that advantage, it has the disadvantage of being inefficient because the initial period is dropped in any case. Moreover, the inefficiency can be very large in the presence of missing data, because first differences can be built only on ensuing observations. For example, if one person is observed at $$t = 1, 3, 5,$$ then FE would use the three person-years, but in FD the person would be dropped completely. With (balanced) panel data for $$T = 2$$, the DiD estimator is identical to FE and FD. For longer panels, however, it differs in general. In fact, it can give misleading answers because all variables enter the regression in levels. If there are control variables (which usually is the case) their effect is likely to be biased, which may also induce bias on the treatment effect. It is therefore recommended to use FE (or FD) where all variables are transformed (Wooldridge, 2010, p. 321).

The advantage of FE in terms of not wasting precious observations should not be underestimated. While things are getting better and nowadays you will often be able to have access to panel time series (panel data with very long $$T$$), in past as well as still in present to a non-trivial extent, most panels have very short $$T$$, in such situations the last thing you want to do is to waste extra observations, and FD has also other issues as mentioned above.

This being said if the level data exhibit very strong correlation you would probably get better results with FD so FD is not useless, but often FE is better more suitable.

• thank you for your answer, +1 for mentioning the equivalence in case of T=2. But I think the rest is completely correct. I have to really on references here, but in "Microeconometrics using Stata" p. 264 Trivedi argues why the FD only seeminly wastes observations. In "Econometric Analysis of Cross-Section and Panel Data" p. 281 Wooldridge argues how, if we are willing to assume strict exogeneity in levels, FD is the most efficient estimator. Both authors state the weak exogeneity assumption in passages close to the cited ones. Commented Jun 14, 2021 at 19:57
• @Papayapap regarding wasting observations I am not sure what Trivedi argument really is here. You can easily verify that FD wastes observation. Since you are using stata, you can run webuse grunfeld this will give you training dataset then you can run FD reg D.(invest mvalue kstock), nocons and you see number of observations utilized is 190, afterwards you can run FE, xtreg invest mvalue kstock, fe and see number of observation jumps to 200. This was example with large T so 10 obs might not be end of the world but with small T the difference would be larger.
– 1muflon1
Commented Jun 14, 2021 at 20:15
• Regarding the strong exogeneity, FD should be strongly exogenous in first differences for FD to be more efficient, I think you either made typo or there is typo in there. Next, they cite weak exogeneity passages saying what? Its possible that under some conditions you can relax strong exogeneity assumption in either FD and FE
– 1muflon1
Commented Jun 14, 2021 at 20:19
• The point is that Trivedi makes is that "Using the LSDV interpretation of the within estimator, the within estimator essentially loses ... observations by estimating the T fixed effects" Commented Jun 14, 2021 at 20:35
• True, I misread that about the strong exogeneity in levels. Commented Jun 14, 2021 at 20:43

Although this conversation is already a bit dated, I want to add a clarification for anyone reading this: FE and FD both effectively use the same amount of information. It can even be shown that a FD-GLS estimator under the usual error components structure is identical to the FE estimator. The fact that Stata reports more observations with the FE than the FD estimator does not accurately reflect this. For the FD estimator, Stata treats the first-differenced observations as "levels". This ignores the fact that the initial level observations are still used to construct the initial differenced observations. Nothing is thrown away. The alternative argument was already mentioned: FE effectively loses the information equivalent to N observations by removing the N within-group means; FD effectively loses the same information by reducing the estimation sample size by 1 per unit.

• Or, in the words of Wooldridge (2010, section 10.7.1): "When we have only two time periods, FE estimation and FD produce identical estimates and inference, as you are asked to show in Problem 10.3" Commented Mar 10, 2023 at 16:46