A researcher uses cross-sectional data to compare wages of job-training participants vs. non-participants. Her colleague argues that people who seek training may already be more motivated — and that this motivation independently raises wages. What does this critique identify?
AA sampling error that can be fixed by collecting a larger cross-sectional sample
BSelection bias: unmeasured stable traits may simultaneously cause training participation and higher wages
CA measurement problem — wages are not accurately captured in cross-sectional surveys
DReverse causation — higher wages lead people to seek training, not the other way around
This is exactly why longitudinal data is needed. If motivated workers both seek training AND earn more, a cross-sectional comparison confounds the training effect with unmeasured motivation. A fixed-effects longitudinal analysis would compare each person's wages before vs. after training, eliminating stable traits like motivation — since they don't change over time — from the comparison. The fundamental issue is time-invariant confounding, not sample size or measurement error.
Question 2 Multiple Choice
A researcher uses a fixed-effects panel model to study the effect of marriage on personal income. Which finding would NOT be estimable with this approach?
AThe average change in income that occurs in the years immediately after marriage
BWhether the income boost from marriage differs by employment sector
CWhether men earn more than women on average
DWhether the income effect of marriage grows or shrinks over time
Fixed effects absorb all time-invariant unit-level characteristics — including sex, which does not change over time for a given person. Sex is perfectly collinear with the unit fixed effect and is therefore inestimable. This is the fundamental tradeoff of fixed-effects modeling: in exchange for eliminating all unmeasured stable confounders, you lose the ability to estimate the effects of any stable characteristic. The approach answers 'what changes when something changes within a person,' not 'how do stable traits relate to outcomes.'
Question 3 True / False
A fixed-effects model eliminates the need to control for any confounding variables when estimating a causal effect from panel data.
TTrue
FFalse
Answer: False
False. Fixed effects only eliminate TIME-INVARIANT confounders — characteristics that remain constant for each unit across the observation period. Time-varying confounders (events that coincide with the treatment and change over time, such as job changes, health shocks, or policy shifts) are not absorbed by the fixed effect and must still be controlled for explicitly. The model controls for stable between-unit differences, not all possible confounding.
Question 4 True / False
Non-random attrition from a longitudinal study can bias effect estimates even when the initial sample was randomly selected from the population.
TTrue
FFalse
Answer: True
True. If participants who drop out of the study differ systematically from those who remain — especially if their likely outcome trajectories differ — then analyses based only on completers will not represent the original population. A study of a medical treatment where sicker patients are more likely to die or withdraw will leave a surviving sample that appears healthier than the true population, biasing estimates of treatment effectiveness. Random initial selection does not protect against this post-randomization bias; it must be addressed through modeling dropout or using inverse probability weighting.
Question 5 Short Answer
What is the key advantage of a fixed-effects model over a cross-sectional regression for causal inference, and what is the cost of that advantage?
Think about your answer, then reveal below.
Model answer: The advantage is that fixed effects eliminate all time-invariant unobserved confounders by comparing each unit to itself across time — the unit-specific intercept absorbs any stable characteristic, measured or not. The cost is that you cannot estimate the effect of any time-invariant predictor (such as sex, race, or country of birth), because these are perfectly collinear with the fixed effects and are differenced out of the estimation.
Cross-sectional regression can only control for observed covariates, leaving unmeasured stable traits as potential confounders. Fixed effects sidestep this by using within-unit variation as the identification strategy — mathematically equivalent to de-meaning each variable by the unit's own time-average before running OLS. The tradeoff defines when fixed effects are appropriate: when the research question concerns within-unit change and when unmeasured stable confounders are the primary threat to causal inference. Questions about stable between-unit differences (e.g., does gender affect wages?) require different designs.