ECMT6002 Chap.9 Fixed Effects Panel Models
Fixed Effects Panel Models
Panel data observe the same units repeatedly, allowing estimators to separate persistent unit differences from within unit change. The fixed effects model permits the time invariant unit effect to be correlated with regressors. Demeaning removes that effect and reveals the variation that identifies slope coefficients.
This chapter also covers first differences, time effects, clustered inference and the inability of the within estimator to identify a coefficient on a time invariant regressor. Pooled OLS mixes comparisons across units with changes inside a unit. If persistent managerial quality, ability or location affects the outcome and is correlated with a regressor, that mixture creates omitted-variable inconsistency.
The within transformation subtracts each unit's time average from every observation. Any constant unit component becomes zero, so the slope is estimated from deviations around each unit's own mean. First differencing also eliminates a constant effect by comparing adjacent periods, but it loses the first observation and changes the disturbance dependence.
With two periods the slope estimates coincide; with longer panels their weighting and efficiency differ. Time fixed effects can remove common shocks shared across units at a date. None of these transformations solves time-varying confounding or feedback from current shocks into future regressors.
Standard errors should respect within-unit dependence, often through clustering at the unit level when the sampling structure supports it. A time-invariant regressor disappears under demeaning because it contains no within variation, which is an identification limitation rather than evidence that the variable is unimportant.
What this chapter covers
- 01
Panel data structure
- 02
Unit effects and correlation
- 03
Within transformation
- 04
First difference comparison
- 05
Time fixed effects
- 06
Cluster robust inference
Worked example · free
Applying the within transformation
- +1The unit mean of x is 4 and the unit mean of y is 7.
- +1Subtract the x mean to get demeaned x values of minus 2 and 2.
- +1Subtract the y mean to get demeaned y values of minus 2 and 2.
- +1Any time invariant unit effect is also minus itself after demeaning and therefore disappears.
Key terms
- Panel data
- Repeated observations on the same cross sectional units over time.
- Unit effect
- A time invariant unobserved component specific to a panel unit.
- Within estimator
- An estimator using deviations from unit means to remove unit fixed effects.
- Clustered standard error
- A covariance estimate allowing arbitrary disturbance dependence within defined clusters under its regularity conditions.
Fixed Effects Panel Models FAQ
Why do time invariant regressors disappear?
Their value equals their unit mean in every period. Subtracting the unit mean therefore creates a column of zeros, leaving no within unit variation from which to estimate a separate slope.
Do fixed effects remove all omitted variables?
No. They remove omitted factors that are constant within a unit over time. Time varying omitted factors can still be correlated with regressors and bias the within estimator.
When should standard errors be clustered by unit?
It is common when disturbances may remain dependent across time within the same unit. The choice must match the sampling and assignment structure, and a small number of clusters requires extra care.
How should I interpret a within estimate?
Describe a change for the same unit relative to its own average, holding included time-varying controls fixed. Do not turn it into a comparison between permanently different units. Then state that time-invariant heterogeneity has been removed while time-varying confounding and feedback may remain, so causal language still depends on the panel exogeneity assumptions.
Exam move
Start every panel problem by drawing the unit and time indexes. Mark each variable as time varying or time invariant, then state whether identification comes from within or between variation. Practise the demeaning algebra on tiny panels and explain what survives. Keep the coefficient estimator question separate from the covariance clustering question.
Construct a two-unit, three-period table and calculate unit means, demeaned values and first differences by hand. Check that each unit's demeaned column sums to zero and that a constant regressor becomes all zeros. Interpret the within slope as a comparison of the same unit at times when x differs from its own average, never as a cross-sectional comparison.
Write one example of time-invariant confounding that fixed effects removes and one time-varying confounder it leaves behind. Add date indicators to the thought experiment and explain which aggregate shock they absorb. For strict exogeneity, trace whether an outcome shock today could change a future regressor; if so, the familiar within consistency claim needs reconsideration.
Choose clustering from the likely dependence and assignment structure rather than as an automatic software option. End by identifying the variation discarded by the estimator and the substantive parameter that can no longer be learned from the transformed data.
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