Mathematics
Fixed Effects (in Panel Data)
Quick fact
Fixed effects can only estimate effects of variables that change over time; any time-invariant factors (like gender or birthplace) are automatically controlled for and cannot be separately estimated.
Why this is interesting
Have you ever wondered how researchers can claim a policy caused a change when they can't run an experiment? Fixed effects gives them a powerful trick: compare each person to themselves over time.
Read the full explanation
Understanding Fixed Effects (in Panel Data)
Imagine you want to know if a new teaching method improves test scores. If you compare different students, many factors (like family income) could bias the result. But if you follow the same students over time, you can see how each student's score changes when the method is introduced. Fixed effects uses this idea for entire datasets: it treats each student (or city, company, etc.) as its own control, removing all permanent differences. Practically, it adds a separate 'dummy variable' for every unit or subtracts each unit's average over time. The result: you only use variation within a unit, not between units.
A deeper explanation
The mechanism works by eliminating unobserved unit-specific heterogeneity that is constant over time. Suppose we model outcome Yit = β Xit + αi + εit, where αi captures all time-invariant features of unit i. With fixed effects, we estimate β using only deviations from each unit's mean (demeaning), which removes αi. This requires the assumption of strict exogeneity: the error term εit is uncorrelated with X at all time periods after controlling for the fixed effects. The method is equivalent to including a dummy variable for each unit (hence 'fixed effects'). Important limitations: it cannot estimate coefficients for time-invariant variables, and it can be inefficient if there is little within-unit variation. Despite this, it is a standard first step in causal analysis because it eliminates a major source of bias—unobserved confounding due to stable characteristics.