Year Country Geography CurrentPolitics ChancellorVisit TradeWithGermany
1 1990 Finland 1500 -0.4738078 152.72004 107.05227
2 1991 Finland 1500 1.7566339 148.09848 97.35936
3 1992 Finland 1500 -1.5531657 147.80263 98.52434
4 1993 Finland 1500 -1.3853555 150.05642 108.38914
5 1994 Finland 1500 -1.3254298 147.58362 102.70535
6 1995 Finland 1500 0.8584380 148.64469 96.46766
7 1990 France 400 -0.9530696 32.12637 25.61469
8 1991 France 400 0.1494820 39.29465 28.65127
9 1992 France 400 -0.3079278 31.96830 25.85378
10 1993 France 400 2.3832139 41.56196 31.36891
11 1994 France 400 -0.5136004 41.46984 20.30912
12 1995 France 400 -0.7170133 41.17091 34.09697Fixed Effects
Will Horne
Panel Data
Panel Data (or longitudinal data) is data in which you observe the same unit over multiple time periods
A panel survey in which the same individual is resurveyed over multiple rounds
Test scores for schools over multiple years
State level unemployment rates
Because the same units appear in the data many times, our OLS assumptions will be violated
- Are independent errors plausible in these cases?
Hierarchical Data
Sometimes, our data is nested in some way that creates a hierarchy
For example, individuals might be nested in cities
Cities are nested in states
And states are nested in countries
Again, if we don’t account for this structure, our estimates will be biased.
Fixed Effects
The most popular (but not the only) way to deal with these data structures is with fixed effects models.
- With nested data, random effects models are often good options. Beyond the scope of this class but see posted reading.
Intuition: Fixed effects models allow us to control for the invariant features of these units. This allows us to estimate within unit effects
Example: International Trade Policy
Imagine we want to know whether a visit from Germany’s chancellor increases trade with Germany in the next year
Global trade data is often structured as “stacked” data, with the each row being a country year, with each country entering the data many times
Trade policy with Germany might also depend on things like the local culture, a countries history with Germany, geography, resource endowments and more.
- This would make a messy DAG, how can we deal with this?
Panel Data Structure
DAG
Adding a Fixed Effect
If we add a “fixed effect” for a country, we control for everything about that country that does not vary (or varies very slowly) over the time period of the data
For example, controlling for “France” implicitly controls for France’s proximity to Germany, it’s resource endowment, it’s political culture and it’s history with Germany
- This makes identifying effects much easier!
Cleaner DAG
What do fixed effects do?
Essentially, fixed effects control for the unit
- Can be: Country, individual, state, school, year, etc
As with any control, this means we hold the unit fixed and looked only at variation within the unit
Key point: We get rid of the “between” variation. Maybe we don’t want to do this!
Pooled regression retains between variation (but is likely biased)
Random effects allow for partial pooling
- We will return to these, but bias concerns remain!
Simple Example
Imagine we tracked two people and looked at how many hours per week they exercised and how many colds they got every year. Data from just two years might look like this:
# A tibble: 4 × 5
# Groups: Individual [2]
Individual Year Exercise MeanExercise WithinExercise
<chr> <dbl> <dbl> <dbl> <dbl>
1 Esnaina 2024 5 6 -1
2 Esnaina 2025 7 6 1
3 Mariam 2024 4 3.5 0.5
4 Mariam 2025 3 3.5 -0.5Pooled Data
What is the relationship between colds and exercise?
Pooled Regression
Adding Individual Means
Think of the individual means as the origin (0,0) on each individual graph
Just Between Variation
These are the individual specific factors that we are going to control away. It appears that Mariam has higher propensity for colds (and excercise)!
Individual Data
Combining and Regressing
Intuition
We removed all of the between variation by finding the individual specific mean of each variable, and then using only that variation
By using fixed effects, we lose the ability to utilize between group variation
- If we are interested in between variation, we cannot use fixed effects!
By controlling for an individual, we are also controlling for all time-invariant features of that individual
We still need to control for things that vary within the individual
Mechanics
The standard way to write a fixed effects regressions is as follows:
\[ Y_{it} = \beta_{i} + \beta_{1}X_{it} + \epsilon_{it} \]
What is different from the standard OLS line?
We now have individual specific intercepts \(\beta_{i}\)
\(X_{it}\) now varies across individuals and time
Our error is generated by both individual and temporal factors. Sometimes the error might be decomposed, for example \(u_{i}\) and \(\epsilon_{t}\) plus any idiosyncratic error
Varying Intercepts
Tip
It may not be immediately obvious, but letting the intercept vary by individual is the same as adding a “dummy” (aka binary) variable for the individual?
What does the coefficient of a binary variable tell us?
Equivalently: a control for “India” tells us the change we would expect in Y, given that we are in India rather than not in India.
Gapminder
Country Specific Intercepts
So, we see that at any level of GDP per Capita, India has a longer life expectancy by ~15 years.
Mechanics
One way to fit fixed effects regression: include a dummy variable for every individual
Or, fit the following regression:
\[ Y_{it} - \bar{Y}_{i} = \beta_{0} + \beta_{1}(X_{it} - \bar{X}_{i}) + \epsilon_{it} \]
Why does this work?
fixest packge
OLS estimation, Dep. Var.: lifeExp
Observations: 1,704
Fixed-effects: country: 142
Standard-errors: Clustered (country)
Estimate Std. Error t value Pr(>|t|)
log(gdpPercap) 9.76896 0.701507 13.9257 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 5.06038 Adj. R2: 0.832467
Within R2: 0.410119Note - there can be problems fitting fixed effects on non-continous data, but fixest has implemented options for most GLMs that we’ve discussed.
Interpretation
OLS estimation, Dep. Var.: lifeExp
Observations: 1,704
Fixed-effects: country: 142
Standard-errors: Clustered (country)
Estimate Std. Error t value Pr(>|t|)
log(gdpPercap) 9.76896 0.701507 13.9257 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 5.06038 Adj. R2: 0.832467
Within R2: 0.410119How should we interpret this?
In a year where the log of GDP per capita is one unit higher than it typically is for that country, we would expect life expectancy to be about 9.8 years longer than it typically is for that country
Policy Specific Example
Imagine we wanted to know whether building more houses leads to increases or decreases in home prices in the following quarter
We can imagine various things we would want to control for, like crime rates, quality of education in the city, etc
But, both new builds and housing prices are probably also driven by unobservable, city specific factors.
We can “control away” those city specific factors (as long as they don’t vary over time in our data) by using fixed effects
The Data
Min. 1st Qu. Median Mean 3rd Qu. Max.
130710 453759 648997 706368 1051994 1339969 city popularity crime edu_quality new_houses price
1 Atlanta -0.9034835 41.07687 6.915975 29 242180.7
2 Atlanta -0.9034835 49.29670 4.428896 33 257568.8
3 Atlanta -0.9034835 44.48307 4.720051 28 213320.5
4 Atlanta -0.9034835 52.05405 5.361506 20 211581.7
5 Atlanta -0.9034835 41.67420 4.830725 31 267976.9
6 Atlanta -0.9034835 39.43562 4.002756 24 263874.4Pooled Data
City Level Averages
This is the “between” relationship. We can interpret this as “cities where more houses are built tend to have higher prices”
Within Effects
Naive (Pooled) Model
Call:
lm(formula = price ~ new_houses + crime + edu_quality, data = sim_data)
Residuals:
Min 1Q Median 3Q Max
-705236 -119945 -17964 118379 671889
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -347899.5 40615.0 -8.566 <2e-16 ***
new_houses 22329.2 402.8 55.437 <2e-16 ***
crime 197.5 506.9 0.390 0.697
edu_quality 1561.0 5314.1 0.294 0.769
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 185000 on 1296 degrees of freedom
Multiple R-squared: 0.7039, Adjusted R-squared: 0.7032
F-statistic: 1027 on 3 and 1296 DF, p-value: < 2.2e-16Fixed Effects Model
OLS estimation, Dep. Var.: price
Observations: 1,300
Fixed-effects: city: 13
Standard-errors: Clustered (city)
Estimate Std. Error t value Pr(>|t|)
new_houses -392.977 160.2970 -2.45156 3.0506e-02 *
crime -403.296 38.2012 -10.55717 1.9874e-07 ***
edu_quality 1958.689 663.3114 2.95290 1.2079e-02 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 30,302.4 Adj. R2: 0.991937
Within R2: 0.027989Summing Up
Why the Naive Model Gets it “Wrong”: Popular cities have higher housing prices and more new homes built. When you pool data across all cities, you capture that correlation.
How Fixed Effects Fix the Bias: Fixed effects allow each city to have its own intercept. The regression then looks within each city’s variation—eliminating city-level confounders. The Within-city estimate on new houses flips to negative (increased supply lowers price.
When to use Fixed Effects: Whenever you suspect time-invariant differences across groups (cities, states, firms, etc.) could bias your estimates.
Assumptions and Cavaets
Each regressor must vary over time for at least some i, and cannot be perfcetly colinear with fixed effects
Fixed Effects cannot solve reverse causality
A model (FE or pooled) of police spending per capita will predict that more police causes more crime. Why?
- Other time series approaches more appropriate, but beyond scope
Cannot solve time variant unobserved heterogeniety
Time invariant variables must have time-invariant effects (see Ren and Allison on Canvas)
Random Effects?
Random effects allow for “partial pooling”
- Rather than giving letting each unit specific mean take on any value, we assume \(\beta_{i}\) follow a known distribution.
Random effects have some nice features
Estimates are more precise (smaller standard errors)
Uses information about all cases to improve estimates of individual intercepts
Rather than within variation, estimates are a weighted average of within and between variation
- Is this a good thing? Maybe!
Random Effects Problems
Random effects only isolate a credible within variation if the individual intercepts are unrelated to our treatment + control variables
- This seems unlikely. Would be saying “The unobserved city effect is not related to new housing builds, crime, education, etc
In practice, the simple random effects model is almost never used anymore
Random effects live on in hierarchical/multilevel modelling. Important and useful, but unfortunately beyond the scope…
Mcelreath’s Statistical Rethinking (2020)or Gelman and Hill (2006) are both good introductions
Should you cluster your standard errors?
Standard errors are supposed to be independent, but if data is clustered this may not hold
Conventional wisdom is to cluster your standard errors at the level of your fixed effects.
fixesteven does this automatically. But….Only necessary if you have treatment effect heterogeniety
or if your treatment assignment is clustered (ie, a whole classroom gets treated)
See Abadie et al paper on canvas for much more!
Two Way Fixed Effects
Can we get greedy and try to control away unobserved variation from multiple sources, like individuals and they city they live in
Yes!, but…
Suppose we do use city and individual. Now we are at only looking within-individual and within-city, controlling all city specific and individual specific heterogeniety
Individuals who don’t move don’t show up in the regression because they don’t have any variation on city, and so their effect is “absorbed”.
Two Way Fixed-Effects with Time
It would be very nice to also be able to adjust for any trends that vary with time
This was a very popular method in papers on development.
Imai and Kim show that generally: “It is impossible to simultaneously and non-parametrically adjust for unit-specific and time-specific unobserved confounders under the two-way fixed effects framework”
- Robert Kubinec has a good blog post on issues with TWFE: https://www.robertkubinec.com/post/fixed_effects/
We will return to these points when we cover Diff-in-Diff