Fixed Effects (Two-Way FE)

Motivating Example: The Effect of Unionization on Wages

You want to know whether joining a union raises a worker's wages. You have panel data tracking thousands of workers over ten years, observing their union status and wages each year. A simple cross-sectional OLS regression of wages on union membership is almost certainly biased: workers who join unions differ systematically from those who do not — in ability, motivation, industry, occupation, and dozens of other ways you cannot fully measure.

But here is the key insight: with panel data, you can compare the same worker before and after they join a union. Whatever unobserved characteristics make worker $i$ different from worker $j$ — ability, motivation, family background — those characteristics are constant over time for the same worker. The fixed effects model strips all of that confounding out.

This within-unit comparison is the power of fixed effects: it exploits within-unit variation over time, effectively asking "what happens to the same worker's wages when their union status changes?" rather than "do workers in unions earn more than workers not in unions?"

A. Overview: The Logic of Fixed Effects

The Panel Data Model

Consider a simple panel model:

where:

• $Y_{it}$ is the outcome (wages) for unit $i$ at time $t$
• $X_{it}$ is the treatment/covariate of interest (union status)
• $\alpha_i$ is a unit fixed effect — everything about unit $i$ that does not change over time (ability, gender, race, permanent personality traits)
• $\lambda_t$ is a time fixed effect — everything that changes over time but affects all units the same way (macroeconomic conditions, inflation, policy changes)
• $\varepsilon_{it}$ is the idiosyncratic error

What Fixed Effects Do

Including $\alpha_i$ (unit fixed effects) is equivalent to including a separate dummy variable for each unit. This inclusion absorbs all time-invariant differences between units — both observed and unobserved.

Including $\lambda_t$ (time fixed effects) absorbs all temporal shocks that are common across units.

Together, two-way fixed effects (TWFE) control for unit-level permanent differences and common time trends. The identifying variation that remains is within-unit, over-time variation that deviates from common trends.

What Fixed Effects Do NOT Do

Common Confusions

B. Identification: The Within Transformation

The Math Behind "Demeaning"

The within transformation works by subtracting the unit-specific mean from each variable. Define:

Subtracting the unit mean from the original equation:

The fixed effect $\alpha_i$ has vanished — because $\alpha_i - \alpha_i = 0$. What remains is the within-unit variation, which is why this approach is called the "within estimator."

The Key Assumption: Strict Exogeneity

For the FE estimator to be consistent, you need strict exogeneity:

This condition means the error at time $t$ is uncorrelated with the regressors at all time periods — past, present, and future. This restriction rules out feedback effects (where past outcomes affect future treatment). If there are lagged dependent variables, strict exogeneity fails, and FE is biased (Nickell bias).

The Hausman Test: FE vs. RE

The Hausman test compares FE and RE estimates. Under the null hypothesis that RE is consistent (i.e., the unit effect $\alpha_i$ is uncorrelated with the regressors), both FE and RE are consistent but RE is more efficient. Under the alternative, only FE is consistent.

A significant test statistic rejects RE in favor of FE.

C. Visual Intuition

Imagine a scatterplot of wages (Y) vs. years of experience (X) for multiple workers. Without fixed effects, you would draw one regression line through all the data. But each worker has a different starting wage (some are high-ability, some low-ability), creating parallel clouds at different heights.

Fixed effects is like drawing a separate regression line for each worker (or equivalently, centering each worker's data on their own mean). You are no longer comparing high-ability workers to low-ability workers. You are looking at how each worker's own wages change when their experience (or union status) changes.

The slope you estimate is the average of these within-person slopes — purged of all between-person differences.

D. Mathematical Derivation

E. Implementation

1library(fixest)
2
3# One-way FE
4fe1 <- feols(wages ~ union + tenure + hours_worked | worker_id,
5           data = df, vcov = ~worker_id)
6summary(fe1)
7
8# Two-way FE
9fe2 <- feols(wages ~ union + tenure + hours_worked | worker_id + year,
10           data = df, vcov = ~worker_id)
11summary(fe2)
12
13# Hausman test
14library(plm)
15pdata <- pdata.frame(df, index = c("worker_id", "year"))
16fe_plm <- plm(wages ~ union + tenure + hours_worked, data = pdata, model = "within")
17re_plm <- plm(wages ~ union + tenure + hours_worked, data = pdata, model = "random")
18phtest(fe_plm, re_plm)

F. Diagnostics and Robustness Checks

Is There Enough Within Variation?

If your key variable barely changes within units over time, FE will have very low power. Check the within variation:

• In Stata: xtsum variable reports between and within standard deviations.
• If the within SD is tiny relative to the between SD, FE is identifying off very little variation, and your estimates will be imprecise.

F-Test for Joint Significance of Fixed Effects

Test whether the unit fixed effects are jointly significant. If not, pooled OLS may be preferred. In Stata: test after areg or check the F-stat from reghdfe.

Cluster Your Standard Errors

Robustness to Functional Form

Report results with and without time fixed effects. If the coefficient changes dramatically when you add time FE, common time trends were confounding your estimate. Also consider adding unit-specific linear trends if you worry about differential trends.

Interpreting Results

• The FE coefficient measures the within-unit effect: "when union status changes for a given worker, their wages change by $\hat{\beta}$ on average."
• This coefficient is not the same as the cross-sectional comparison: "workers in unions earn $\hat{\beta}$ more than workers not in unions."
• If the FE estimate is much smaller than the OLS estimate, it suggests that positive selection (higher-ability workers select into unions) was inflating the OLS coefficient.
• Time fixed effects control for common shocks. If wages grew nationally during your sample period, time FE prevent you from attributing that growth to changes in union status.

G. What Can Go Wrong

H. Practice

I. Swap-In: When to Use Something Else

• Random Effects: When you believe unobserved unit effects are uncorrelated with regressors (Hausman test does not reject). More efficient than FE and can estimate time-invariant effects.
• Correlated Random Effects (Mundlak): Add group means of time-varying regressors to the RE model. Gives FE-equivalent coefficients for time-varying variables while also estimating time-invariant effects.
• First differencing: Instead of demeaning, take first differences: $\Delta Y_{it} = \beta \Delta X_{it} + \Delta \varepsilon_{it}$. Equivalent to FE with two periods; with more periods, FE is generally more efficient unless errors follow a random walk.
• Arellano-Bond GMM: For dynamic panels (lagged dependent variable) where FE is biased. Uses past levels as instruments for first-differenced equations.

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