A researcher uses panel data on workers with individual fixed effects to estimate the wage premium of earning a college degree. A critic argues the estimate may still be biased. What is the most plausible reason?
AFixed effects models cannot be validly applied to wage data
BFixed effects already absorb all possible confounders, so no bias can remain
CA time-varying confounder — such as a worker's expanding professional network that simultaneously drives degree completion and wage growth — is not eliminated by fixed effects
DThe panel has too many time periods, inflating standard errors and biasing coefficients
Fixed effects eliminates bias from time-invariant confounders (things stable about a person — innate ability, family background), but time-varying confounders remain a problem. If something that changes over time simultaneously affects both the treatment variable (degree completion) and the outcome (wages), fixed effects cannot remove that bias. This is the most common misconception about panel data: many students believe fixed effects 'solves endogeneity' without this important caveat.
Question 2 Multiple Choice
Why is the fixed effects (within) estimator less efficient than OLS applied to pooled cross-sectional data?
AFixed effects uses both between and within variation, creating overidentification
BFixed effects uses only within-unit variation over time, discarding the between-unit variation that pooled OLS exploits
CFixed effects requires the random effects assumption, which is typically violated
DFixed effects cannot control for more than one regressor simultaneously
The within estimator identifies causal effects by comparing each unit to itself over time, which means it uses only within-unit variation and ignores all between-unit differences. Pooled OLS uses both between and within variation. When the random effects assumption holds (individual effects uncorrelated with regressors), discarding the between variation is wasteful — random effects is more efficient. Fixed effects is the right choice when you cannot trust between-unit comparisons due to omitted variable bias, not because it extracts more information.
Question 3 True / False
Fixed effects estimation is algebraically equivalent to applying OLS to data demeaned at the individual level, because subtracting each unit's time-average eliminates the unit-specific fixed effect.
TTrue
FFalse
Answer: True
The individual fixed effect α_i is constant over time, so when you subtract each unit's time-mean from every observation, α_i cancels: (y_it − ȳ_i) = (x_it − x̄_i)'β + (u_it − ū_i). The demeaned regression has no individual-specific intercept to estimate, and OLS on this demeaned data gives the same coefficient estimates as the within estimator. This is why fixed effects is sometimes called the 'within estimator' — it exploits only within-unit variation.
Question 4 True / False
A panel with more time periods (larger T) typically produces more precise estimates than a panel with fewer time periods, regardless of the variation in the data.
TTrue
FFalse
Answer: False
Precision depends on how much within-unit variation exists in the treatment variable, not just on T. If a variable barely changes within units over time (e.g., a country's constitution rarely changes), adding more time periods adds little useful variation. Identification in panel data comes from units that actually change their treatment status — if nothing changes, more periods don't help. The optimal dimension (more units vs. more time) depends on where the relevant variation exists.
Question 5 Short Answer
Why does observing the same unit over multiple time periods give panel data an advantage over cross-sectional data for causal inference?
Think about your answer, then reveal below.
Model answer: Cross-sectional data compares different units, so any stable difference between them — in ability, background, or unmeasured characteristics — can confound the estimated relationship. Panel data compares each unit to itself at different points in time. Because stable characteristics are constant within a unit, they cancel out when we look at within-unit changes. The causal question shifts from 'do units with the treatment differ from units without it?' to 'does the same unit change when its treatment status changes?' — a much cleaner comparison that eliminates bias from all time-invariant confounders.
The union membership example from the topic illustrates this concretely: cross-sectional estimates of union wage premia are inflated because high-ability workers disproportionately join unions. Panel estimates controlling for worker fixed effects are much smaller, because comparing the same worker before and after union membership holds ability constant.