Panel Data and Fixed Effects, on One Page

Panel data (a.k.a. longitudinal data) is what you have when you follow the same units — people, firms, countries — across several time periods, instead of a single snapshot of many different units. That repetition buys you something a plain cross-section can't: a way to erase every stable trait of a unit, whether you measured it or not.

Here's the problem it solves. Suppose each unit i has some fixed, unobserved characteristic α_i — a firm's management culture, a person's baseline ability — that shapes y and correlates with your regressor X. That's textbook endogeneity: α_i sits in the error term, and pooled OLS across all units confuses "units with high X" for "units that were going to have high y anyway." The estimate can be badly off — even point the wrong way (a Simpson's paradox).

The within-transformation is the trick. Write the model with that nuisance in it, one intercept per unit:

$y_{it} = \alpha_i + \beta X_{it} + \varepsilon_{it}$

Then, for each unit, subtract that unit's own average over time from every variable:

$y_{it} - \bar{y}_i = \beta\,(X_{it} - \bar{X}_i) + (\varepsilon_{it} - \bar{\varepsilon}_i)$

Anything constant within a unit — including the whole nuisance $\alpha_i$ — equals its own mean, so it cancels. What's left is pure within-unit variation: how y moves when X moves for the same unit. The slope of that is the fixed-effects estimate of β. You've controlled for every time-invariant confounder without ever naming one.

That is a statement about the stable confounder only. For the slope to be causal, the changes in X within a unit still have to be exogenous — formally $E[\varepsilon_{it} \mid X_{i1}, \ldots, X_{iT}, \alpha_i] = 0$, the strict exogeneity the estimator leans on.

Three things to carry away:

• The cost: fixed effects burn all between-unit variation, so anything that never changes for a unit (someone's sex, a country's coastline) drops out — you simply can't estimate its effect this way.
• Standard errors need care. Observations from the same unit are usually correlated across time, and conventional standard errors can understate uncertainty badly when they are. Cluster-robust standard errors, clustered by unit, account for that dependence — with their own caveat that they need a decent number of clusters to behave.
• Two-way and beyond. Add time fixed effects to absorb shocks common to all units in a period. That machinery leads into difference-in-differences, though it does not by itself make a DiD: that design needs parallel trends for the untreated outcome. And when treatment arrives at different times for different units, the standard two-way estimate can mislead — with heterogeneous effects it quietly uses already-treated units as controls and weights the comparisons badly.

Fixed effects defeat time-invariant confounding, not endogeneity in general. A confounder that moves within a unit and tracks X, reverse causality, feedback from past outcomes — each still biases the estimate, and each needs something more: the confounder measured and controlled for, time effects for common shocks, a design built around the timing, or an instrument. The full treatment — within vs first-differences, random effects, clustered SEs, staggered DiD — is in the Panel Data (Fixed Effects) note.