Pain-aware intro (and where to find the “why” pages)
If you’re staring at an econometrics problem set at 11:47pm, you’re not alone: fixed effects (FE) questions are designed to feel “almost obvious” right up until you try to write the argument cleanly. On exams, the most common failure mode is not algebra—it’s explaining what FE is actually doing and why it changes the bias story.
If you’re building your conceptual foundation across topics, start at our Why Hub. If you want the broader econometrics & time series roadmap, see Econometrics & Time Series.
Answer first
Fixed effects reduces omitted variable bias when the omitted confounder is time-invariant within each unit (e.g., a person’s baseline ability, a firm’s culture, a county’s geography). FE works by removing each unit’s constant component from the data (via “demeaning” or differencing), so any unobserved factor that is constant over time cannot explain within-unit variation and therefore cannot bias the within-unit estimate.
Problem setup
Consider a standard panel model with unit \(i\) observed over time \(t\):
\[ y_{it} = \beta x_{it} + \alpha_i + u_{it}. \]
Here:
- \(y_{it}\) is the outcome,
- \(x_{it}\) is a regressor that changes over time,
- \(\alpha_i\) is an unobserved, time-invariant unit effect (the “omitted variable” that lives inside the unit),
- \(u_{it}\) is the idiosyncratic error.
The key issue: if \(x_{it}\) is correlated with \(\alpha_i\), pooled OLS is generally biased because \(\alpha_i\) is in the error term from the pooled perspective.
Step-by-step solution (WordPress-safe MathJax)
Step 1: Define the within-unit time average
Let the time average for unit \(i\) be:
\[ \bar{y}_i = \frac{1}{T}\sum_{t=1}^{T} y_{it}, \quad \bar{x}_i = \frac{1}{T}\sum_{t=1}^{T} x_{it}, \quad \bar{u}_i = \frac{1}{T}\sum_{t=1}^{T} u_{it}. \]
Taking time averages of the model:
\[ \bar{y}_i = \beta \bar{x}_i + \alpha_i + \bar{u}_i. \]
Step 2: Subtract the average equation from the original
Subtract the averaged equation from the original equation:
\[ y_{it} – \bar{y}_i = \beta (x_{it} – \bar{x}_i) + (\alpha_i – \alpha_i) + (u_{it} – \bar{u}_i). \]
The fixed effect \(\alpha_i\) cancels:
\[ \tilde{y}_{it} = \beta \tilde{x}_{it} + \tilde{u}_{it}, \]
where \(\tilde{y}_{it} = y_{it} – \bar{y}_i\), \(\tilde{x}_{it} = x_{it} – \bar{x}_i\), and \(\tilde{u}_{it} = u_{it} – \bar{u}_i\).
Step 3: See why the “omitted variable bias” channel is removed
Pooled OLS bias comes from treating \(\alpha_i\) as part of the error and allowing \(\mathrm{Cov}(x_{it}, \alpha_i) \neq 0\). But FE estimates \(\beta\) using only within-unit variation \(\tilde{x}_{it}\). Since \(\alpha_i\) is constant within unit, it has no within-unit variation and cannot confound \(\tilde{x}_{it}\) once it is removed.
Step 4: State the identifying assumption clearly
FE does not require \(\mathrm{Cov}(x_{it}, \alpha_i) = 0\). Instead, it relies on a within-unit exogeneity condition such as:
\[ \mathbb{E}(u_{it}\mid x_{i1},\dots,x_{iT},\alpha_i)=0. \]
Informally: after controlling for unit fixed differences, the remaining shocks \(u_{it}\) are not systematically related to the regressors (often phrased as “strict exogeneity”).
Intuition
FE is like asking: “When this same unit changes its \(x\), does its \(y\) change?” Any stable, unit-level trait (ability, baseline demand, permanent productivity) cannot explain changes within the unit over time. So FE protects you from confounding by anything that doesn’t move over time inside the unit.
Common exam mistakes
- Claiming FE fixes all endogeneity. It only handles time-invariant confounding. Time-varying omitted variables can still bias FE.
- Forgetting the identifying assumption. Instructors often want you to state strict exogeneity (or a weaker assumption for specific settings).
- Confusing FE with first differences. They are related but not identical in finite samples; they coincide under specific conditions (e.g., \(T=2\)).
- Interpreting \(\beta\) as cross-sectional. FE is a within-unit effect, not a between-unit comparison.
Why this matters
In applied economics, FE is one of the most common ways to make causal claims using observational panel data. If you can explain—cleanly—what gets removed, what variation remains, and what assumption you still need, you’ll be able to handle panel questions, DiD intuition, and many applied empirical papers with much less confusion.
Final summary
- FE reduces omitted variable bias by removing unit-specific, time-invariant confounders \(\alpha_i\).
- It estimates \(\beta\) from within-unit variation \((x_{it}-\bar{x}_i)\).
- You still need an exogeneity condition on the remaining error \(\tilde{u}_{it}\) (often strict exogeneity).
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