2. Instruments and Panel Data

Panel Fixed Effects

Use within-unit variation while recognising time-varying confounding and limited support

Panel Fixed Effects

Panel data change the comparison

For unit $i$ and time $t$ ,

Y_{it}=\tau D_{it}+\alpha_i+\lambda_t+\varepsilon_{it}.

$\alpha_i$ absorbs time-invariant unit differences;
$\lambda_t$ absorbs shocks common to all units in a period;
$\tau$ is estimated from within-unit treatment changes after common time changes are removed.

Units that never change $D$ do not directly identify $\tau$ , although they can help estimate other components.

The within transformation

With unit fixed effects only, subtract each unit’s time mean:

Y_{it}-\bar Y_i=\tau(D_{it}-\bar D_i)+(\varepsilon_{it}-\bar\varepsilon_i).

The unobserved constant $\alpha_i$ disappears. A permanent characteristic such as birthplace also disappears, so its coefficient cannot be separately estimated in the standard within regression.

For two-way fixed effects, subtract unit and time means and add the grand mean.

Compare pooled and two-way within slopes

The pooled slope mixes between-unit differences with within-unit change. The two-way transformation removes the constructed unit and period components; real data require a causal argument for the remaining change.

What fixed effects do not remove

Threat	Example
time-varying confounding	a local recession triggers policy and changes employment
anticipation	firms adjust before the clean-air zone starts
reverse causality	rising pollution accelerates policy adoption
measurement error	differencing amplifies noisy treatment changes
dynamic response	current outcome depends on past policy and past outcome
spillovers	traffic moves from treated to comparison municipalities

“Controls for all unit characteristics” means all time-invariant additive unit components in the specified model—not every unobserved factor.

Strict exogeneity is demanding

A standard fixed-effects argument often requires

E[\varepsilon_{it}\mid D_{i1},\ldots,D_{iT},\alpha_i]=0.

Current shocks must not predict past or future treatment after conditioning. Policy adopted in response to an outcome shock can violate this condition even when unit fixed effects are present.

Variation and weighting audit

Report:

how many units switch treatment and when;
within versus between variation;
units lost because outcomes or treatment do not vary;
leverage of unusual switchers;
clustering and serial-correlation treatment;
whether the coefficient averages heterogeneous effects with problematic weights.

For causal interpretation of two-way fixed-effects regressions, Imai and Kim (2021) emphasise explicit counterfactual and treatment-history assumptions rather than a purely mechanical panel specification.

Quick check

A municipality’s policy changes once, immediately after a pollution spike. Do municipality and year fixed effects identify the policy effect?

Answer

Not automatically. The time-varying spike may both trigger adoption and predict future pollution. Fixed effects remove stable municipality differences and common year shocks, not this municipality-specific dynamic confounding.

Next: Dynamic Panels and Limits

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Weak Instruments and LATE

Diagnose unstable IV ratios, weak-identification inference and complier-specific interpretation

Dynamic Panels and Limits

Handle lagged outcomes, anticipation, persistence and unbalanced observation without overclaiming