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Week 8 · Dr. Aluma Dembo

Fixed Effects in Panel Data

Fixed Effects in Panel Data

Part of: Econometrics Lecture 08 — Applied Econometrics, Dr. Aluma Dembo Key concepts: Panel Data, Fixed Effects, Within Estimator, Demeaning, Between Variation, Within Variation, Time Fixed Effects, Two-Way Fixed Effects, Clustered Standard Errors, Individual Fixed Effect, First Stage, Two Stage Least Squares

What is Panel Data?

Panel data tracks the same individuals (people, firms, cities, judges…) across multiple time periods.

  • Each individual appears in the dataset more than once — once per time period
  • This gives you both cross-sectional variation (differences between individuals) and time variation (changes within an individual over time)
Why panel data is powerful

With cross-sectional data, you can only compare different individuals to each other. With panel data, you can compare the same individual to themselves at different points in time — which controls for all the stable, individual-level characteristics you can't observe or measure.

Motivating Example: Temperature and Immigration Judges

Paper: Heyes & Saberian (2019)

Question: Does outdoor temperature affect how immigration judges make decisions?

Why ask this? If judges are supposed to be impartial, then outdoor temperature — which is outside their control — should have zero effect on their decisions. A significant temperature effect would suggest that physiological or psychological factors are influencing what should be a purely legal judgement.

Data:

  • 206,924 asylum decisions
  • 266 immigration judges
  • 43 cities
  • January 2000 – September 2004

Outcome (ritr_{it}): whether the judge granted asylum (1 = yes, 0 = no), as a fraction across decisions in a given period

Treatment (hith_{it}): outdoor temperature (°F) on the day of the hearing

Building Up the Model: Step by Step

Step 0: The Naive Model

rit=α+βhit+uitr_{it} = \alpha + \beta h_{it} + u_{it}

Just regress asylum grant rate on temperature. Problem: this ignores that asylum grant rates differ enormously across nationalities (some countries have much higher grant rates than others), and across cities (some courts are more liberal).

Step 1: Control for Nationality

rit=α+βhit+γ⋅nationalityit+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality}_{it} + u_{it}

After adding nationality fixed effects, the coefficient on temperature is marginally insignificant (p=0.059p = 0.059).

Why add nationality controls?

Nationality strongly predicts grant rate — Syrian or Afghan asylum seekers have much higher grant rates than others. If certain nationalities cluster in certain cities, and those cities happen to be hotter, nationality would confound the temperature-grant rate relationship. We control for it to remove this bias.

Time Fixed Effects: Controlling for Calendar Shocks

Any event that affects all judges in all cities on the same day is a "time shock". Examples:

  • A big Supreme Court ruling that changes interpretation of asylum law
  • A major news event that shifts public opinion

Time fixed effects absorb these: one dummy variable per time period that captures everything that changes uniformly across all observations in that period.

Repeated Cross Sections with Time Controls

We can add time controls in two ways:

Option A — Linear time trend: include a variable t=1,2,3,…t = 1, 2, 3, \ldots (day, month, or year number)

rit=α+βhit+γ⋅nationality+δ1day_of_week+δ2month+δ3year+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality} + \delta_1 \text{day\_of\_week} + \delta_2 \text{month} + \delta_3 \text{year} + u_{it}

Option B — Dummy variable time fixed effects: one dummy per time period (more flexible — allows any non-linear time pattern)

rit=α+βhit+γ⋅nationality+∑τλτDτ+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality} + \sum_{\tau} \lambda_\tau D_\tau + u_{it}
Linear trend vs. dummy time fixed effects

A linear time trend forces the relationship with time to be a straight line. Dummy time fixed effects make no such assumption — they let the time effect be completely flexible, zig-zagging in any pattern. Dummy FEs use more degrees of freedom but are more robust to non-linear time patterns.

What does a time fixed effect "absorb"?

Imagine a nationwide policy change that makes judges tougher in a particular month. A month fixed effect absorbs this — it picks up the average shift in grant rates in that month across all cities. After including it, the remaining variation in grant rates is within a given month, across cities.

Between Variation vs. Within Variation

This distinction is central to understanding fixed effects.

Between Variation: Comparing Averages Across Individuals

rˉi=1T∑t=1Trithˉi=1T∑t=1Thit\bar{r}_i = \frac{1}{T} \sum_{t=1}^T r_{it} \qquad \bar{h}_i = \frac{1}{T} \sum_{t=1}^T h_{it}

Between variation: the variation in mean grant rates across judges / cities.

Why between variation is dangerous for causal inference

If some cities are both hotter and more lenient (e.g. because of the local political culture), then comparing city averages will show a correlation between temperature and leniency — but it's driven by the city culture, not temperature. This is between-individual confounding (also called self-selection or omitted variable bias).

Example: Alfie lives in Tel Aviv (hot and strict courts) while Barak lives in Seattle (cold and lenient courts). Comparing them tells you about Tel Aviv vs. Seattle culture, not about temperature effects.

type: fixed-effects
mode: within-between
The sign can flip entirely

Each coloured cluster is one individual (judge/city), and within every cluster the slope is negative — hotter days mean a lower grant rate. But the cluster means (black X's) climb from bottom-left to top-right, so pooled OLS reports a positive between-slope. That dashed line is pure confounding: hotter places happen to be stricter for unrelated reasons. Fixed effects throw away the between variation and keep only the within slopes — recovering the true negative effect.

Within Variation: Comparing the Same Individual Across Time

r¨it=rit−rˉih¨it=hit−hˉi\ddot{r}_{it} = r_{it} - \bar{r}_i \qquad \ddot{h}_{it} = h_{it} - \bar{h}_i

Within variation: how much a variable deviates from that individual's own mean.

r¨it\ddot{r}_{it} asks: "On this day, did judge ii grant asylum more or less than she usually does?" h¨it\ddot{h}_{it} asks: "Was today hotter or cooler than this city's average temperature?"

The power of within variation

By looking only at deviations from individual means, you control for everything that is stable about that individual — their personality, their background, the court culture, the city's climate on average. Within variation compares Alfie to Alfie on a hot day vs. a cool day, not Alfie to Barak.

The Individual Fixed Effects Model

yit=β0+β1xit+ai+uit\boxed{y_{it} = \beta_0 + \beta_1 x_{it} + a_i + u_{it}}
  • yity_{it}: outcome for individual ii at time tt
  • xitx_{it}: time-varying treatment/covariate
  • aia_i: individual fixed effect — a catch-all for everything about individual ii that doesn't change over time
  • uitu_{it}: remaining idiosyncratic error
What does aia_i absorb?

aia_i captures all time-invariant characteristics of individual ii — observed or unobserved. For a judge, this might include:

  • Personality traits (inherent strictness/leniency)
  • Legal training and philosophy
  • Regional court culture
  • Average climate of their city

You never need to measure these things — the fixed effect absorbs them all. This is why fixed effects are so powerful for causal inference.

  • β1\beta_1 measures the average within-individual effect of xx on yy: "when xx increases for individual ii, how does yy change for that same individual ii?"

How to Estimate Fixed Effects: Within-Estimation (Demeaning)

You could, in principle, add a dummy variable for every individual ii in the regression. But with many individuals (e.g. 266 judges), this is unwieldy. Within-estimation (demeaning) achieves the same result more efficiently.

Step 1: Demean all variables

For each individual ii, compute their time-average, then subtract it:

y¨it=yit−yˉix¨it=xit−xˉi\ddot{y}_{it} = y_{it} - \bar{y}_i \qquad \ddot{x}_{it} = x_{it} - \bar{x}_i

This removes the individual fixed effect aia_i entirely (since aˉi=ai\bar{a}_i = a_i for a time-constant term):

y¨it=β1x¨it+u¨it\ddot{y}_{it} = \beta_1 \ddot{x}_{it} + \ddot{u}_{it}

Step 2: Run OLS on the demeaned data

β^1within=∑i∑tx¨ity¨it∑i∑tx¨it2\hat{\beta}_1^{\text{within}} = \frac{\sum_i \sum_t \ddot{x}_{it} \ddot{y}_{it}}{\sum_i \sum_t \ddot{x}_{it}^2}
Demeaning in practice — judge example

Judge Smith makes decisions on 200 cases. Her average grant rate is 35%. On hot days, her grant rate is 30%; on cool days, 40%.

After demeaning:

  • Hot days: r¨=30%−35%=−5%\ddot{r} = 30\% - 35\% = -5\%
  • Cool days: r¨=40%−35%=+5%\ddot{r} = 40\% - 35\% = +5\%

We regress these deviations on deviations of temperature from Smith's average temperature. If hotter-than-usual days consistently correspond to lower-than-usual grant rates, β^1<0\hat{\beta}_1 < 0.

type: fixed-effects
mode: demeaning
What demeaning does to the data

On the left, each individual sits at its own height — the X's mark their personal means (xˉi,yˉi)(\bar x_i, \bar y_i). Subtracting those means slides every cluster onto a common origin (right), erasing the individual fixed effect aia_i. All that remains is the within variation, and a single regression through the stacked clusters recovers the within slope β^1\hat\beta_1 — identical to adding a dummy for every individual, but far cheaper.

Adding individual dummies vs. demeaning: same result

Running OLS with a dummy for every individual ii produces exactly the same β^1\hat{\beta}_1 as demeaning. The demeaning approach just avoids including hundreds of dummy variables explicitly. This equivalence is the Frisch-Waugh-Lovell theorem in action again.

City Fixed Effects

Apply individual fixed effects at the city level:

rit=α+βhit+γ⋅nationality+αc+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality} + \alpha_c + u_{it}

where αc\alpha_c is a fixed effect for city cc — absorbing all time-invariant city characteristics (court culture, demographics, local legal norms, average climate).

r
feols(res ~ Temperature + region | city, data = asy.data)

Result: After adding city fixed effects, the temperature coefficient becomes insignificant (p=0.54p = 0.54).

What this tells us

The raw (unadjusted) relationship between temperature and grant rate is explained by city-level confounders — not by temperature itself. Hotter cities may simply have stricter courts for unrelated reasons. Once we control for city fixed effects, the within-city variation in temperature is unrelated to grant rates.

Two-Way Fixed Effects: City + Month

Control for both city-level and month-level fixed effects simultaneously:

rit=α+βhit+γ⋅nationality+αc+αm+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality} + \alpha_c + \alpha_m + u_{it}
r
feols(res ~ Temperature + region | city + month, data = asy.data)
  • αc\alpha_c: absorbs city characteristics
  • αm\alpha_m: absorbs any systematic variation in grant rates across months (e.g. summer vs. winter court backlogs)
Why add month fixed effects?

Temperature is seasonal — it's higher in summer months than winter months. If courts are also systematically busier or stricter in summer (e.g. because immigration flows spike), then without month FEs, you'd be comparing "hot summer days in city X" to "cold winter days in city X" — confounded by the summer/winter court dynamic. Month FEs remove this confound.

Interacted Fixed Effects: City × Month

Go further: a fixed effect for each city-month combination:

rit=α+βhit+γ⋅nationality+αcm+uitr_{it} = \alpha + \beta h_{it} + \gamma \cdot \text{nationality} + \alpha_{cm} + u_{it}
r
feols(res ~ Temperature + region | city^month, data = asy.data)
  • 43 cities × ~12 months = ~463 fixed effects
  • Absorbs anything that is stable within a given city-month cell (e.g. "Boston courts in January always behave this way")
  • More demanding — uses more degrees of freedom but provides much cleaner identification

Judge × Month Two-Way Fixed Effects

The most demanding specification: a fixed effect for each judge-month combination:

rit=α+βhit+αjm+uitr_{it} = \alpha + \beta h_{it} + \alpha_{jm} + u_{it}
r
feols(res ~ Temperature | judge^month, data = asy.data)
  • 266 judges × ~12 months ≈ 2,677 fixed effects
  • Absorbs individual judge tendencies that vary by month (e.g. Judge Smith is always stricter in January)
  • Highest within-R2R^2: this specification explains the most variance in grant rates
Progression of fixed effect specifications
Specification Fixed effects What's controlled
Nationality only Nationality Applicant country
+ City FE Nationality + City Everything stable about a city
+ Month FE Nationality + City + Month Stable city traits + seasonal patterns
City × Month FE Nationality + City×Month City traits + city-specific seasonality
Judge × Month FE Judge×Month Everything stable about each judge each month

Clustered Standard Errors

When using fixed effects, the standard OLS standard errors are usually wrong. Why? Because observations within the same group (same judge, same city) are likely correlated with each other in ways the model doesn't fully capture.

What are clustered standard errors?

They account for within-group correlation of errors — the fact that observations in the same cluster are not independent.

Cluster at the level of the fixed effect\text{Cluster at the level of the fixed effect}
Rule of thumb

Cluster your standard errors at the level of your fixed effect (or coarser). If you have city fixed effects, cluster at the city level. If you have judge fixed effects, cluster at the judge level. This ensures your inference is valid even when errors are correlated within cities/judges.

Comparison of SE types

SE type When to use Code
Huber-White (HC) Heteroskedasticity across observations (different error variance for different obs) vcov = 'hetero'
Newey-West (HAC) Time series: serial correlation + heteroskedasticity vcov = 'newey_west'
Clustered Panel data: within-group error correlation cluster = ~city
r
# Heteroskedasticity-robust SEs
feols(res ~ Temperature | city, data = asy.data, vcov = 'hetero')

# Newey-West HAC SEs
feols(res ~ Temperature | city, data = asy.data, vcov = 'newey_west')

# City-clustered SEs
feols(res ~ Temperature | city, data = asy.data, cluster = ~city)
Don't use default (homoskedastic) SEs with panel data

Default OLS standard errors assume all errors are i.i.d. — equal variance and independent. In panel data with fixed effects, within-group error correlation is almost guaranteed. Using default SEs will make your estimates look more precise than they are, inflating t-statistics and producing spuriously significant results.

Preview: Instrumental Variables in Panel Data

Paper: Cai, De Janvry & Sadoulet (2013) — insurance take-up among Chinese rice farmers.

Research question: Does social learning / peer effects drive insurance take-up? (i.e. if your friends buy insurance, does that make you buy it too?)

The endogeneity problem: A farmer's insurance take-up might be correlated with their friends' take-up simply because they face similar risks (shared weather, shared information) — not because of peer influence. This is correlated unobservables — a classic OVB problem.

The Instrument: Default Buy Option

In a randomised trial, some farmers were assigned a default "buy" option (opt-out rather than opt-in for insurance). This randomisation:

  1. Affects their friends' actual take-up rate (relevant ✅)
  2. Only affects the farmer's own take-up via how it affects their friends' take-up (excludable ✅) — the default assignment was random, so it has no direct effect on the farmer's fundamentals
First stage: pre_takeup_rateit=γ0+γ1default_buyit+ηit\text{First stage: } \text{pre\_takeup\_rate}_{it} = \gamma_0 + \gamma_1 \text{default\_buy}_{it} + \eta_{it}
F-stat=54.8(strong instrument)F\text{-stat} = 54.8 \quad \text{(strong instrument)}

2SLS in feols

r
feols(takeup ~ 1 | pre_takeup_rate ~ default_buy, data = farmers)

The | separates the first-stage instrument from the main equation. The F-stat of 54.8 far exceeds the rule-of-thumb threshold of 10, confirming a strong first stage.

Wu-Hausman test for endogeneity

The Wu-Hausman test checks whether the endogenous regressor is actually endogenous (i.e. whether OLS is biased). Null hypothesis: OLS is consistent (regressor is exogenous). If the test rejects, we have evidence that IV is necessary.

A significant Wu-Hausman test here confirms that friends' take-up rate is genuinely endogenous — OLS would have given biased estimates.

Why does randomisation of default options solve endogeneity?

Default assignment is random — farmers didn't self-select into "default buy" vs. "default don't buy". So it's uncorrelated with the farmer's own risk exposure, risk preferences, and other confounders. The only channel through which default assignment can affect a farmer's outcome is via their peers' take-up. This is exactly what makes it a valid instrument.

Summary

  1. Panel data tracks the same individuals over time. It enables within-individual comparisons that control for all time-invariant characteristics.

  2. Between variation compares averages across individuals — vulnerable to cross-individual confounders (self-selection, culture, etc.).

  3. Within variation compares an individual to themselves over time — controls for all stable individual traits, observed or not.

  4. Individual fixed effects model: yit=β0+β1xit+ai+uity_{it} = \beta_0 + \beta_1 x_{it} + a_i + u_{it}. The aia_i absorbs all time-invariant traits of individual ii. β1\beta_1 is the within-individual effect of xx.

  5. Within-estimation (demeaning): subtract each variable's individual mean (y¨it=yit−yˉi\ddot{y}_{it} = y_{it} - \bar{y}_i), then run OLS on the demeaned data. Equivalent to adding individual dummy variables.

  6. City fixed effects control for time-invariant city characteristics. After adding them, temperature's effect on asylum grant rates disappears — city confounders were driving the raw correlation.

  7. Two-way fixed effects add a second dimension (e.g. city + month, or judge × month) for more rigorous control of multiple sources of confounding.

  8. Clustered standard errors are required when observations within a group (city, judge) have correlated errors. Cluster at the level of the fixed effect.

  9. IV in panel data: when the key regressor is endogenous even within individuals, use instruments. The default buy randomisation isolates exogenous variation in peers' take-up, enabling causal identification of peer effects.