A researcher uses panel data to estimate the wage return to education with unit (person) fixed effects. She then tries to include a variable for each person's race. What happens?
AThe race variable gets a small but significant coefficient because FE controls for confounding
BThe race variable is perfectly collinear with the person fixed effects and drops out of the regression entirely
CThe FE estimator becomes inconsistent because race is endogenous
DThe coefficient on education becomes biased toward zero because race absorbs some of its variation
Race is a time-invariant characteristic — it doesn't change within a person over time. Fixed effects work by demeaning each person's data, which removes everything that is constant within that person. A time-invariant variable like race is constant within each person, so after demeaning it becomes a column of zeros — perfectly collinear with the unit dummies. This is a fundamental limitation of FE: you cannot estimate the level effect of any variable that has no within-unit variation.
Question 2 Multiple Choice
A researcher studies how investment affects firm productivity using a firm fixed effects model. She argues: 'Since firms that are already highly productive tend to invest more, this within-firm correlation makes FE estimates biased.' Is she correct?
ANo — FE removes all endogeneity by controlling for firm-level unobservables
BNo — within-firm variation is by definition exogenous because firm identity is held constant
CYes — FE only removes time-invariant bias; if productivity shocks cause within-firm investment changes, FE estimates are still biased
DYes — FE should not be used when the outcome (productivity) causes the regressor (investment)
The researcher is correct. Fixed effects removes bias from time-invariant unobserved confounders (like a firm's permanent management quality), but it does not remove bias from time-varying confounders. If a firm experiences a positive productivity shock and responds by increasing investment within the same period, the within-firm variation in investment is correlated with the within-firm error. FE cannot solve this — it is a specific solution to a specific problem, not a general cure for endogeneity.
Question 3 True / False
The within transformation (subtracting each unit's time-mean from every observation) eliminates time-invariant unobserved heterogeneity because any characteristic that doesn't change over time becomes zero after demeaning.
TTrue
FFalse
Answer: True
This is the core insight behind fixed effects. If a unit's unobserved characteristic α_i is constant over time, then subtracting the unit's mean (ȳ_i = α_i + other terms) from each observation removes α_i exactly. What remains is only within-unit variation over time. This is mathematically equivalent to including a dummy variable for every unit — the dummies soak up the permanent unit-level differences.
Question 4 True / False
Fixed effects models eliminate most forms of omitted variable bias in panel data, which is why they are the preferred estimator whenever panel data is available.
TTrue
FFalse
Answer: False
Fixed effects eliminates only time-invariant omitted variable bias. If unobserved confounders change over time within units — for example, if workers who receive wage increases are simultaneously experiencing changes in unobserved motivation — FE estimates remain biased. Two-way FE additionally controls for common time shocks, but time-varying unit-level confounders still require additional strategies (instrumental variables, DiD, etc.). Choosing FE over random effects also involves tradeoffs in efficiency and the Hausman test.
Question 5 Short Answer
Explain why a fixed effects regression cannot estimate the coefficient on a time-invariant variable such as gender or country of origin.
Think about your answer, then reveal below.
Model answer: Fixed effects works by demeaning each unit's observations — subtracting the unit's time-average from every period's value. A time-invariant variable like gender has the same value in every period for a given unit, so after demeaning it becomes identically zero for all observations. A column of zeros carries no information and is perfectly collinear with the unit fixed effects (which are also constant within each unit). The estimator literally cannot distinguish the effect of gender from the unit's permanent fixed effect — they are mathematically inseparable.
The intuition is that FE only uses variation within units over time. Since gender never varies within a person, there is no within-unit variation to exploit. This is the price of the FE strategy: you gain protection against time-invariant confounders, but you lose the ability to estimate time-invariant effects. If estimating the effect of time-invariant variables is the goal, you need a different strategy (random effects or between estimator), with the corresponding tradeoffs in bias and efficiency.