Recall the fundamental problem: omitted variables that are correlated with X bias our estimates. In cross-section, we can only control for what we observe.

Panel data gives us a different strategy: if the omitted variable does not change over time for a given unit (ability, culture, geography), we can remove it entirely by exploiting within-unit variation.

We are no longer comparing different people to each other. We are comparing the same person to themselves at different points in time.

Y_it = β₀ + β₁X_it + α_i + ε_it

i indexes units, t indexes time. α_i is the unobserved individual effect, it varies across units but is fixed within a unit over time.

α_i captures everything about unit i that is constant over time: innate ability, culture, geography, management quality.

The key question: is α_i correlated with X_it? The answer determines which estimator to use.

Fixed effects uses only within variation. If X barely changes within units over time, FE will be imprecise, there is little variation left to identify β₁.

Stack all NT observations and run OLS as if they were independent. This treats α_i as part of the error term.

If Cov(α_i, X_it) ≠ 0, pooled OLS is biased, exactly the omitted variable problem from cross-section, now with a time dimension.

Even if α_i is uncorrelated with X, pooled OLS standard errors are wrong because observations from the same unit are correlated over time.

Verdict: pooled OLS is almost never the right choice with panel data.

Take the model in period t and subtract period t − 1:

ΔY_it = β₁ΔX_it + Δε_it

α_i is time-invariant, so it cancels out exactly. We are left with changes in Y regressed on changes in X.

First differencing is efficient when T = 2. For longer panels, the fixed effects estimator is generally preferred.

Subtract each unit’s time mean from both sides:

(Y_it − Ȳ_i) = β₁(X_it − X̄_i) + (ε_it − ε̄_i)

α_i is absorbed by the unit mean and drops out. Running OLS on the demeaned data gives the FE estimator.

Equivalently: include a dummy variable for each unit. The FE estimator is identical to OLS with N − 1 unit dummies. This is the least squares dummy variable (LSDV) estimator.

Degrees of freedom: we lose N − 1 for the unit dummies (or equivalently, for the demeaning).

If Cov(α_i, X_it) = 0, the unobserved effect is just a component of the composite error. We do not need to remove it, we can model it.

The RE estimator uses both within and between variation. It is a weighted average of the FE (within) estimator and the between estimator. This makes it more efficient than FE when its assumption holds.

RE can also estimate coefficients on time-invariant regressors (race, sex, geography), something FE cannot do.

But if Cov(α_i, X_it) ≠ 0, RE is biased. The assumption is often implausible in economic data.

Both FE and RE are consistent if Cov(α_i, X_it) = 0. Only FE is consistent if it is not.

H₀: RE is consistent, the unobserved effect is uncorrelated with the regressors. If H₀ holds, FE and RE estimates should be similar.

The test statistic is based on the difference in coefficient vectors: H = (β̂_FE − β̂_RE)′ [Var(β̂_FE) − Var(β̂_RE)]⁻¹ (β̂_FE − β̂_RE), distributed χ²_k.

A significant Hausman statistic rejects RE in favor of FE. In most applied economics work, FE is the default.

DiD compares the change over time in a treated group to the change over time in a control group.

β̂_DiD = (Ȳ_treat,post − Ȳ_treat,pre) − (Ȳ_ctrl,post − Ȳ_ctrl,pre)

This is equivalent to a two-way FE regression with unit and time dummies and a treatment indicator D_it = 1 if unit i is treated in period t.

Key assumption: parallel trends. In the absence of treatment, treated and control units would have followed the same trend. This cannot be tested directly but can be examined with pre-treatment data.

Observations from the same unit over time are correlated, not independent. Using standard (or even HC robust) SEs that assume independence will understate uncertainty.

Cluster-robust SEs at the unit level allow for arbitrary serial correlation within units. They are the standard in applied panel work.

If treatment varies at a higher level (e.g., state policy applied to individuals), cluster at the treatment assignment level, not the observation level.

With few clusters (< 30–50), cluster-robust SEs can be unreliable. Consider wild bootstrap or other small-sample corrections.