Lecture 4 - Controls and Fixed Effects
Topics in Econometrics - M2 ENS Lyon
2025-09-24
Outline of the course
Introduction and Fundamentals
Simulations
Design: Identification and Beyond
Controls and Fixed Effects
IV and RDD
Modelling
Analysis
Communication
Goals of the session
Controls: what do controls do to our regression?
Fixed effects are extremely common in applied economics
What are they really doing?
More generally, what are we really estimating in a specific model?
What are we comparing to what?
Where does the identifying variation come from?
Controls and projections
Regression as a projection
Frisch–Waugh–Lovell (FWL) Theorem
\[Y = X\beta + W\delta + U\]
- The estimate of \(\beta\) is the same as the estimate of \(\tilde{\beta}\) in:
\[Y^{\perp W} = X^{\perp W}\tilde{\beta} + U^{\perp W}\]
where \(.^{\perp W}\) denotes each variable where \(W\) has been residualized
ie its projection onto the orthogonal space to W
Obtained using:
- The projection matrix \(P_W = W(W'W)^{-1}W'\)
- The residual-maker matrix \(M_W = I - P_W\)
eg \(X^{\perp W} = M_W X\)
Fixed effects regression = regression on variables after partialling out the fixed effects
In practice
To compute the partialled out version of a regression:
- Compute the residualized version of \(y\) and \(x\): regress them on controls/FE
- Regress the residuals on one another
Exercise. Using the data bellow, run two regressions and compare the estimates obtained:
- Regress
l_murderonl_pris,male,populationandincome - Regress their residualized versions of
l_murderandl_prison one another (partialling outmale,populationandincome)
- Regress
Visualizing the raw data
# Graph levels
guns |>
ggplot(aes(x = prisoners, y = murder)) +
geom_point() +
labs(
title = "Relationship between incarceration and murder rates",
subtitle = "Variables in level: need to transform it",
x = "Incarceration rate",
y = "Murder rate"
)
# Graph logs
guns |>
ggplot(aes(x = l_pris, y = l_murder)) +
geom_point() +
geom_smooth(method = "lm") +
labs(
title = "Relationship between incarceration and murder rates",
subtitle = "Log are better suited",
x = "Log of incarceration rate",
y = "Log of murder rate"
)Illustration of the FWL theorem
#demeaning and showing that equal to residuals
guns_demean <- guns |>
mutate(
l_murder_res = lm(data = guns, formula = l_murder ~ male + population + income) |>
residuals(),
l_pris_res = lm(data = guns, formula = l_pris ~ male + population + income) |>
residuals()
)
reg <- guns |>
lm(formula = l_murder ~ l_pris + male + population + income) |>
broom::tidy() |>
mutate(reg = "raw", .before = 1)
reg_res <- guns_demean |>
lm(formula = l_murder_res ~ l_pris_res - 1) |>
broom::tidy() |>
mutate(reg = "residualized", .before = 1)
rbind(reg, reg_res) |>
filter(str_starts(term, "l_pris")) |>
kable()| reg | term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|---|
| raw | l_pris | 0.8702664 | 0.0251012 | 34.67037 | 0 |
| residualized | l_pris_res | 0.8702664 | 0.0250583 | 34.72968 | 0 |
Why does that matter?
In economics, we are typically interested in one coefficient, that of the treatment
The true relationship of interest is the partialled out one
Can bring back a complex linear model to a bivariate regression
What matters is the variation in the treatment after partialling out
Where does the variation in the treatment comes from?
Interpretation of the coefficient of interest: “Comparing …”
ATE as a weighted average
The estimate of the treatment coefficient is in fact a weighted average of individual treatment effects \(\tau_i\)
Weight: \(w_{i} = (T_{i} - \mathbb{E}[T_{i} | X_{i}])^{2}\)
The weight represents:
How well the controls explain the treatment status
The conditional variance of the treatment, given \(X_i\)
Actually equivalent to leverage in the residualized regression
Implications
Observations whose treatment status is largely explained by covariates therefore contribute little, if at all, to estimation
For FE: if for some groups there is little within variation, these groups do not contribute to identification
Implications for external validity and representativity
May weigh more some characteristics
Implications for statistical power: the effective sample might be much smaller than the nominal sample
If need a CIA, weights are not equal
Effective sample vs nominal sample
Figure from Aronow and Samii (2016)
Two types of controls
Can distinguish controls between those that are identification-related and those that are not
Identification-related controls are those that are necessary for the CIA to hold
Other controls are here to improve precision of the estimator
- Explain \(y\) \(\Rightarrow\) \(\searrow \ \sigma_u^{2}\) (and \(\nearrow \ R^2\)) and therefore \(\searrow \ \mathbb{V}_{\hat{\beta}}\)
Identification based on repeated observations
Adjusting for non-varying factors
Repeated observations over some dimension allow adjusting for all the unobserved characteristics that are constant across that dimension
Transform each variable into its deviation from the group mean
Only keep within variation (discards the between)
Two approaches to do that:
- Manual demeaning
- Including fixed effects
Basically build a counterfactual
Event studies, DiD, and TWFEs
Objective: estimate the impact of some treatment at a certain time
Leverages repeated observations, typically panel data
Builds a counterfactual that can be explicit or more implicit (eg TWFE):
- Unit’s outcome had the event not occurred
Event study
Potentially, all units are treated
Assumed counterfactual: group’s past value
Within variation only
Flexible, allows looking at whether effects are dynamic
Difficult to rule out other things changing at the same time
- The rooster concluding the sun rises because of his crowing?
\[Y_{it} = \sum_{t = -K}^{\tau - 2} [\beta_t \mathbb{1}\{t\}] + \beta_{\tau} \mathbb{1}\{\tau\} + \sum_{t = \tau + 1}^{L} [\beta_t \mathbb{1}\{t\}] + e_{it}\]
DiD, DiDiD, TWFE
- Some units never get treated
- Assumed counterfactual: parallel trends of treated and untreated are parallel
- Within and between variation
- Pre-trends not a problem (unlike event studies) as long as trends of the groups are parallel
- Issues when go beyond simple binary DiD (we discuss that later)
\[Y_{it} = \beta G_{i}P_t + \lambda_G + \lambda_P + e_{it}\]
Nuts and bolts of fixed effects
Interpreting fixed effects
Group FEs: compare individuals within the group
Time FEs: compare individuals within a time period
TWFEs:
Average of TEs identified from variation within group and variation within period
\(\neq\) variation within “that group that year” (this would be group-year FEs)
Including FEs changes the estimand: we compare observation within a group or within a time period
Equivalence residual vs manual demean
#demeaning and showing that equal to residuals
sample_demean <- guns |>
mutate(
l_murder_res = feols(data = guns, fml = l_murder ~ 1 | state) |>
residuals()
) |>
group_by(state) |>
mutate(mean_l_murder = mean(l_murder)) |>
ungroup() |>
mutate(
l_murder_demean = l_murder - mean_l_murder
) |>
select(l_murder_res, l_murder_demean) |>
head(10) | l_murder_res | l_murder_demean |
|---|---|
| 0.2824963 | 0.2824963 |
| 0.2170183 | 0.2170183 |
| 0.2094711 | 0.2094711 |
| 0.2094711 | 0.2094711 |
| 0.1057927 | 0.1057927 |
| -0.0098917 | -0.0098917 |
| -0.1515422 | -0.1515422 |
| -0.1300360 | -0.1300360 |
| -0.0883633 | -0.0883633 |
| -0.0582103 | -0.0582103 |
FWL with fixed effects
library(fixest)
#demeaning and showing that equal to residuals
guns_demean <- guns |>
mutate(
l_murder_res = feols(data = guns, fml = l_murder ~ 1 | state) |>
residuals(),
l_pris_res = feols(data = guns, fml = l_pris ~ 1 | state) |>
residuals()
)
reg_fe <- guns |>
fixest::feols(fml = l_murder ~ l_pris | state, cluster = "state") |>
broom::tidy() |>
mutate(reg = "fixed_effects", .before = 1)
reg_res <- guns_demean |>
feols(fml = l_murder_res ~ l_pris_res - 1, cluster = "state") |>
broom::tidy() |>
mutate(reg = "residualized", .before = 1)
rbind(reg_fe, reg_res) |>
kable()| reg | term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|---|
| fixed_effects | l_pris | -0.15834 | 0.0365294 | -4.334587 | 7.05e-05 |
| residualized | l_pris_res | -0.15834 | 0.0365138 | -4.336438 | 7.01e-05 |
Playing around
Let’s run some R code together to understand how fixed effects work
Let’s stil will use the
gunsdatasetLet’s consider several regressions, with various sets of fixed effects
Build graphs and interpret the coefficients
FE and Identifying Variation
When adding FE (or controlling in general), we partial out or absorb some of the variation
We throw out variation
Good if throw out variation that:
- Is endogenous
- Explains some of the variance of \(y\) \(\left(\text{since }\mathbb{V}_{\hat{\beta}} = \dfrac{\sigma_u^2}{n \sigma_x^2} \right)\)
Bad if throw out identifying variation, ie variation that allows you to identify the effect of interest
Implicit weighting
Implicit weighting
Each unit does not contribute equally: there are many implicit weights
At least two different objects can be weighted:
Outcomes: weighted difference in mean outcomes
Treatment effects: weighted mean of treatment effects
Often important when effects are heterogenous
Also important: which comparisons are used (ie who counts as “control”)
Aronow and Samii (2016) weights
Weighting outcomes
- Different angle: write \(\hat{\beta}\) directly as a weighted sum of outcomes, not of \(\tau_i\)
\[\hat{\beta} = \sum_i \hat{w}_i \, Y_i\]
Chattopadhyay and Zubizarreta (2023): characterizes these implied weights of linear regression (FE or not)
Need not be non-negative, nor look like the weights of a randomized experiment
\(\Rightarrow\) possible extrapolation: \(\hat{\beta}\) may not represent any real population
Useful as a diagnostic: inspect the weights before trusting \(\hat{\beta}\) (
lmwpackage)
TWFE and 2x2 comparisons
When treatment staggered, TWFE \(\hat{\beta}\) is weighted average of many pairwise DiDs (Goodman-Bacon 2021)
Each compares a newly treated group to a “control”:
Never-treated units, or
Not-yet-treated units
Issue: some “controls” are already-treated units
- Their own treatment effect gets implicitly subtracted out
Negative weights
- These “forbidden comparisons” can assign some \(ATT_{g,t}\) a negative weight
\[\hat{\beta}^{TWFE} = \sum_{g,t} w_{gt} \; ATT_{g,t}, \quad \sum_{g,t} w_{gt} = 1, \quad w_{gt} \ \text{can be} < 0\]
\(\Rightarrow\) \(\hat{\beta}^{TWFE}\) can have the wrong sign
A simple simulation easily illustrates this
Especially problematic when effects vary in time or are heterogeneous
New estimators reweight to recover an interpretable, convex average
de Chaisemartin and D’Haultfœuille (2023)
Can anyone summarize the idea?
What do you get out of it?
What did you think of it?
de Chaisemartin and D’Haultfœuille (2023)
TWFE \(\hat{\beta}\) = weighted sum of \(ATT\)s, weights can be negative
Arises in many standard staggered-adoption designs
Reviews for binary/discrete/continuous treatments, staggered or not:
- Diagnostics: how much of \(\hat{\beta}\) comes from negative weights
- Heterogeneity-robust estimators
Revisits Wolfers (2006) on divorce laws: conclusions change across estimators
Calibrating simulations
Why calibrating?
So far, we considered very simple simulations, with “naive” distributions
Calibrating can help make simulations more realistic
But simulations will never be truly realistic
Yet can still allow to run some sort of robustness check on the ability of your design to retrieve the effects of interest
Also allows you to think about the DGP, your identification strategy, and so on
Fake data simulations
Distributions of the variables
- Emulate the distribution of variables in existing data sets
Fake data simulations
Relationships between variables
Read the literature
Get a sense of typical effect sizes and of relationships between variables
Make assumptions on those relationships. Acknowledge them.
Complexify later if needed. You choose when you stop.
Varying parameters values might change the distribution of some “variables”
eg of the error term
Difficult to work ceteris paribus
Real data simulations
General approach
Start from an existing data set
Not yours. At least not the subset you are interested in
Try to pick a subset where there is not already a treatment effect
Define a treatment allocation mechanism
Add an artificial treatment effect to the outcome variable in your initial data set, eg
\[Y_i(1) = Y_i(0) + \beta_i T_i\]
- Run your analysis and try to recover it
Real data simulations
Complexifying
There is only one artificial aspect in such simulations: the treatment
We can play on only 2 components:
Who is treated? Treatment allocation
Everyone
Only a subset of the population
How? Treatment effect
Homogenous
Heterogenous but random
Some specific correlation structure
Summary
Summary
Today we reviewed:
- The projection/geometric interpretation of regression
- Identification strategies based on repeated observations
- How fixed effects work, under the hood
- Implicit weighting: of outcomes, of treatment effects
- Issues with TWFE under heterogeneous effects
Hopefully you have a better understanding of:
- Causal inference, from a bird’s view
- How fixed effects really work
- Many details and intuitions
Take away messages
The choice of FE is crucial and affects the estimand
FE can remove a lot of variation:
- Great if removes endogenous variation
- Problematic if there is too little variation left
Regressions (with or without FE) implicitly weight units
With heterogeneous or dynamic effects, plain TWFE can be misleading