Suppose your data come in groups — repeated observations of the same people, firms, or districts — and you worry that each group has fixed, unobserved traits correlated with your regressor. Fixed effects (FE) handle this by letting each group serve as its own control.
Write the grouped model with a group-specific intercept:
y_ig = x_ig'β + α_g + ε_igThe within (FE) estimator demeans each variable by its group mean, ỹ_ig = y_ig − ȳ_g, which cancels α_g entirely:
β̂_FE = ( Σ x̃_ig x̃_ig' )⁻¹ Σ x̃_ig ỹ_igAlgebraically it is identical to OLS with a full set of group dummies (LSDV). The payoff is large: FE removes all time-invariant group confounders — observed or not — as long as they are constant within the group. That is a whole class of omitted-variable bias gone without having to measure it.
The price is just as important to understand:
- FE identifies only from within-group variation. Any regressor that is constant within a group is swept away with
α_g— you cannot estimate its coefficient at all. - It discards the between-group variation. If the between-group differences are exactly what you care about, FE throws out your signal. It is then the wrong tool — or at best a robustness check that will attenuate the coefficient you care about.
Where FE sits among the alternatives:
| estimator | assumption on α_g |
uses |
|---|---|---|
| pooled OLS | uncorrelated with x, and zero |
between + within |
| random effects | uncorrelated with x |
between + within (efficient) |
| fixed effects | may correlate with x |
within only |
A Hausman test compares FE and RE: a significant result says α_g is correlated with the regressors, so prefer FE.
And the boundary of what FE can do: it only absorbs confounders that are constant within the group. Anything that varies within the group and still correlates with x — a time-varying shock, say — is untouched, and you are back to needing an instrument or an explicit research design.
Common slips: trying to estimate the effect of a within-group-constant variable under FE (impossible); and reaching for FE when the between-group contrast is the actual object of study.