Mathematics
Fixed Effects
Quick fact
Fixed effects models are often implemented by simply including a dummy variable for each entity (e.g., each person, each company), but mathematically, they are equivalent to subtracting each entity's average over time before analyzing the data.
Why this is interesting
You've likely heard that correlation doesn't imply causation. What if you could remove all the 'background noise' that stays constant over time, getting closer to a cause? That's exactly what fixed effects do.
Read the full explanation
Understanding Fixed Effects
Imagine you're studying whether a new teaching method improves test scores. Different classes have different teachers, different student backgrounds, and different resources—factors that likely affect scores but are hard to measure. With fixed effects, you compare each class to itself over time. You look at how test scores change within a class when the method is introduced, ignoring all the constant differences between classes. This way, any unchanging characteristic of a class (like teacher quality or school culture) cannot bias your results. Fixed effects essentially put each entity under a microscope, focusing only on changes.
A deeper explanation
The fixed effects mechanism works by decomposing the error term. In a panel regression, the unobserved factors that affect the outcome are split into two parts: those that vary over time and those that are constant for each entity. Fixed effects absorb the constant part using dummy variables for each entity or by 'demeaning' the data—subtracting each entity's mean over time from all its observations. This removes all between-entity variation, leaving only within-entity variation. The result is that any omitted variable that is constant over time (like genetics, location, or corporate culture) cannot bias the coefficient of your time-varying explanatory variable. This method is crucial in economics, political science, and sociology, where researchers often have panel data and seek to control for unobserved traits that could otherwise create spurious correlations.