Questions: Within Estimator (Fixed Effects) for Panel Data

Question 1

A researcher studies the effect of R&D spending on firm profits using panel data. She worries about management quality — a stable, unobserved firm characteristic correlated with both R&D decisions and profits. Which feature of the within estimator addresses this concern?

• A. It weights observations by firm size, reducing the influence of high-management-quality outliers
• B. It uses variation in average R&D spending across firms, holding time trends constant
• C. It demeans each firm's observations, removing any stable unobserved characteristic from the estimation
• D. It includes industry dummies, which proxy for sector-level differences in management quality

Answer: (C)

The within estimator subtracts each firm's own time-mean from all its observations. Any stable characteristic — like management quality that doesn't change over the study period — is constant across time for a given firm, so it is completely wiped out by demeaning. The remaining variation is purely within-firm over time, where management quality is held constant by construction. This is the within estimator's core virtue: it controls for time-invariant unobservables without needing to measure them.

Question 2

The within estimator and the between estimator both use the same source of variation in the panel data.

• A. True — both use variation in X across units to identify the effect of X on Y
• B. False — the within estimator uses within-unit variation over time; the between estimator uses across-unit variation in unit averages
• C. False — the within estimator uses variation across time periods; the between estimator uses variation across industries
• D. True — both use the same demeaned data, but apply different weighting schemes

Answer: (B)

The within estimator identifies coefficients from how a given unit's outcome changes when its own regressor changes over time — pure within-unit variation. The between estimator identifies from differences in time-averaged outcomes across units — pure across-unit variation. These are genuinely different sources of variation, and they estimate the same population coefficient only if the model is correctly specified. The within estimator discards all cross-unit information; the between estimator discards all within-unit dynamics.

Question 3

The within estimator eliminates omitted variable bias from most sources of confounding, whether or not the confounders change over time.

• T. True
• F. False

Answer: False

The within estimator only eliminates bias from time-invariant confounders. If an unobserved variable changes over time and correlates with the regressor, the within estimator does not remove that bias. For example, if a firm improves its management in the same year it increases R&D spending, the within estimator cannot separate the R&D effect from the management improvement. Strict exogeneity — requiring errors to be uncorrelated with regressors in all periods — rules out such time-varying confounders.

Question 4

The within estimator requires strict exogeneity: the error at time t must be uncorrelated with the regressors in all periods, not just the current period.

• T. True
• F. False

Answer: True

This is stronger than the contemporaneous exogeneity assumed in cross-sectional OLS. Demeaning creates the transformed error (εᵢₜ − ε̄ᵢ), which includes the average error across all time periods for unit i. If last period's outcome affects this period's regressor (a feedback effect), then the regressor in one period is correlated with errors from other periods — violating strict exogeneity. This rules out dynamic models where lagged outcomes appear as regressors, which is why methods like Arellano-Bond GMM are needed in those settings.

Question 5

Explain intuitively why the within estimator can control for an unobserved variable like 'innate worker ability' even though it never appears in the dataset.

Model answer: If innate ability doesn't change over time, it is identical across all observations for a given worker. When we subtract each worker's own time-mean from all their observations, the ability term — being constant — subtracts out completely. We are left with only within-worker variation over time, in which ability is effectively held constant. We compare each worker to their own past self, which automatically controls for everything stable about them — observed or not.

This is the within estimator's deepest intuition: each unit serves as its own control group. By focusing on changes within a unit rather than differences across units, we hold constant every time-invariant characteristic of that unit, whether or not we can measure it. The only confounders that survive are those that change over time — and strict exogeneity rules those out by assumption.