How can we separate firms that were always more productive from firms that increased training spending?
Panel data follow the same firms, people, schools, or countries over time. This lets us distinguish persistent differences between units from changes that occur within the same unit.
A simple pooled regression may confuse firms that are always more productive with the effect of a firm increasing its own training spending.
Key question
When a given firm changes its training spending, how does its own productivity change?
What makes panel data different?
| Source of variation | Example |
|---|---|
| Between units | some firms consistently spend more on training than others |
| Within a unit | the same firm increases or decreases training spending over time |
| Over time | economy-wide shocks affect all firms in a given year |
Entities across time
yᵢₜThe same entities are observed again at every period.
Fixed effects
withinCentring each entity on its own mean removes whatever never changes about it.
Within vs between
two slopesComparing entities and following one entity can point in opposite directions.
What does fixed effects remove?
Firm culture, location, or long-standing management quality may be unobserved but nearly constant over time. Fixed effects compare each firm with itself after removing its stable average level.
Within-firm interpretation
A training coefficient describes how productivity changes when the same firm changes training spending, net of stable firm-specific characteristics and any included time effects.
Not automatic causality
Fixed effects do not remove omitted factors that change over time, reverse causality, measurement error, or anticipation effects.
What does random effects assume?
Random effects assume the unobserved unit-specific component is uncorrelated with the predictors. When that assumption fails, the coefficient can mix hidden unit differences with the predictor relationship.
| Model | Main source of identification | Key concern |
|---|---|---|
| Fixed effects | changes within the same unit | cannot estimate coefficients for time-invariant predictors |
| Random effects | within- and between-unit variation | requires no correlation between unit effects and predictors |
Within-unit versus between-unit effects
Firms that spend more on training on average may also be better managed. That between-firm relationship can be positive even if increasing training within a particular firm has a smaller effect.
Time effects and clustered uncertainty
Year fixed effects can absorb common shocks. Standard errors are often clustered by unit because repeated observations from the same unit are related.
Change hidden unit differences
Increase the correlation between a firm’s hidden baseline and its predictor. Compare pooled, fixed-effects, and random-effects estimates.
Pooled OLS slope
5.06
Fixed-effects slope
2.00
Random-effects slope
2.72
Entity A
t=1 · x=1 · y=22.0
t=2 · x=2 · y=24.0
t=3 · x=3 · y=26.0
t=4 · x=4 · y=28.0
Entity B
t=1 · x=2 · y=36.0
t=2 · x=3 · y=38.0
t=3 · x=4 · y=40.0
t=4 · x=5 · y=42.0
Entity C
t=1 · x=3 · y=50.0
t=2 · x=4 · y=52.0
t=3 · x=5 · y=54.0
t=4 · x=6 · y=56.0
These slopes are illustrative summaries designed to show how pooled, fixed-effects, and random-effects estimates respond to the simulated data structure.
What to watch
As hidden unit differences become more related to the predictor, pooled and random-effects estimates can move away from the within-unit relationship.