2. Instruments and Panel Data

Dynamic Panels and Limits

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

Dynamic Panels and Limits

Persistence changes the model

Air pollution, earnings and firm productivity are persistent. A dynamic panel may write

Y_{it}=\rho Y_{i,t-1}+\tau D_{it}+\alpha_i+\lambda_t+\varepsilon_{it}.

After within transformation, the demeaned lagged outcome is correlated with the demeaned error in a short panel. This is the source of Nickell bias; adding a lag to a fixed-effects regression is not innocuous.

First differences move, not remove, the problem

Differencing gives

\Delta Y_{it}=\rho\Delta Y_{i,t-1}+\tau\Delta D_{it} +\Delta\lambda_t+\Delta\varepsilon_{it}.

$\alpha_i$ disappears, but $\Delta Y_{i,t-1}$ contains $\varepsilon_{i,t-1}$ and is correlated with $\Delta\varepsilon_{it}$ . Arellano–Bond-style methods use deeper lags as instruments under restrictions on serial correlation and initial conditions.

Instrument proliferation can overfit endogenous variables and weaken specification tests. Report instrument count, lag choices and sensitivity; more instruments are not automatically better.

Dynamic causal effects are not one coefficient

A clean-air policy can have:

anticipation before formal adoption;
an immediate traffic response;
gradual fleet replacement;
displacement to neighbouring areas;
persistence after reversal.

Define treatment history $\bar D_{it}$ and the horizon-specific contrast. A contemporaneous coefficient $\tau$ may mix these processes or impose a constant effect that the policy question does not need.

Lagged outcome: confounder, mediator or state?

Setting	Role of $Y_{t-1}$	Risk
treatment begins after baseline	precision/confounding control	functional-form and overlap
treatment already operated at $t-1$	post-treatment mediator	blocks part of cumulative effect
policy responds to $Y_{t-1}$	treatment assignment predictor	sequential-confounding problem
outcome measured with error	noisy state	differencing can amplify noise

Draw the timeline before deciding to “control for the lag.”

Unbalanced panels and attrition

Units may leave because firms close, pupils migrate or records fail. Fixed effects do not correct selective observation.

Audit:

entry and exit by treatment history;
outcome availability at every intended horizon;
whether treatment changes observation;
balanced-panel and inverse-probability sensitivity;
bounds under plausible missing outcomes.

Restricting to a balanced panel can itself condition on survival.

Choose the simplest design that answers the question

Question	Useful starting point
remove stable unit levels	fixed effects or first differences
estimate immediate policy adoption effect	event-time or DID design with assumptions
model state dependence	dynamic panel with justified lag instruments/model
estimate treatment histories	longitudinal causal methods
forecast future outcomes	predictive panel model, with no automatic causal claim

Quick check

A five-year panel includes lagged earnings and worker fixed effects. Why might the treatment coefficient still be biased?

Answer

The transformed lag is correlated with transformed errors in short panels; treatment may respond to past shocks; lagged earnings may be post-treatment; attrition and measurement error may remain. The exact threat depends on the timeline and target effect.

Next: Policy Evaluation Designs

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Panel Fixed Effects

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

3. Policy Evaluation Designs

Construct untreated outcomes from trends, thresholds, comparable units and weighted donor paths