Fixed Effect Estimation

Beyond Binary Treatments: Fixed Effects Estimators and their Properties

Vladislav Morozov

Contents

• Introduction
• Fixed Effect Estimation
• Causal Properties with General Treatment
• Empirical Application
• Recap and Conclusions

Introduction

Lecture Info

Learning Outcomes

This lecture is about handling more general treatments in panel data using “fixed effect/random intercepts” estimators

By the end, you should be able to

• Describe the fixed effect estimation procedure
• Establish causal properties of such estimators under homogeneous and heterogeneous effects

References

Textbooks:

• Chapter 16 in Huntington-Klein (2025)
• Chapter 13, 14-1, 14-4, 14-5 in Wooldridge (2020)
• Chapter 17 in Hansen (2022) (except dynamic panels and random effects)

Empirical Motivation

Empirical Question

How strongly does pollution affect labor market outcomes?

• We know that pollution is bad for health
• But how does it affect economic activity, particularly earnings and employment?

Challenge: Endogeneity

Cannot just regress labor market outcomes on overall pollution

• Two-way causality, more economically active places tend to have more pollution
• Simple regression will suffer from endogeneity

• Can solve endogeneity with instrumental variables
• But those are difficult to find

Another Approach

• Find pollution not driven by (your own) economic activity
• But some places may be more likely to have this pollution \(\Rightarrow\) this would affect decisions of people to live there

What if we could control for this likelihood?

• How — topic of lecture
• Application: how Borgschulte, Molitor, and Zou (2024) solve the issue

Motivation and Questions

Reminder: TWFE

Recall: for difference-in-differences showed that \[ \small \widehat{ATT}^{DiD} = \hat{\delta} \] where \(\hat{\delta}\) was the OLS estimator in regression \[\small Y_{i2}- Y_{i1} = \gamma + \delta D_{it} + U_{i2} \tag{1}\] for \(\delta\) — the ATT; \(\gamma\) — average change in outcomes (trend)

Reminder: More General Equation

Equation 1 obtained by differencing the two-way fixed effect equation: \[ Y_{it} = \alpha_i + \gamma_t + \delta D_{it} + U_{it}, \tag{2}\] where \[ \gamma_1 = 0, \quad \gamma_2 = \gamma \] and \(\alpha_i = Y_{i1}^0\) — baseline differences between units

Reminder: Estimation

• We know how to apply OLS based on Equation 1: just regress \((Y_{i2}-Y_{i1})\) on \((1, D_{it})\) with OLS
• Last time said that can also apply OLS based on Equation 2 directly (e.g. PanelOLS from linearmodels)
  • Treated \(\alpha_i\) as parameters
  • Got exactly the same results from two approaches

Lecture Questions

1. How to apply OLS on Equation 2?

• What are the regressors? How to “treat \(\alpha_i\) as parameters”?
• How does it work in practice?
• Is the estimator inspired by Equation 2 always equal to the one based on Equation 1?

2. Can we apply the same approach with general (not just 0/1) treatment? What are the causal properties?

Fixed Effect Estimation

Random Intercept (Fixed Effect) Models

First Goal: Vector-Matrix Representation

First question: “treating \(\alpha_i\) as parameters”?

For now forget about \(D_{it}\) and \(\gamma\) and consider: \[ Y_{it} = \alpha_i + U_{it}, \quad i=1,\dots, N; t=1, \dots, T \tag{3}\] \(\alpha_i\) — individual-specific intercept (“unit fixed effect”). Data assumed balanced (same \(T\) for all units)

Want to represent Equation 3 in vector-matrix form

Vector-Matrix Forms for Panel Data I

Before that: more info on matrix forms for panel data.

Vector form as before: single observation (now fixed \(i\) and \(t\)) with vector of covariates: \[ Y_{it} = \bX_{it}'\bbeta + U_{it} \]

Vector-Matrix Forms for Panel Data II

Two key matrix forms:

• Individual level. Let \(\bY_i = (Y_{i1}, \dots, Y_{iT})\), \(\bX_i = (\bX_{i1}, \dots, \bX_{iT})'\), then \[\small \bY_i = \bX_i\bbeta + \bU_i \]

• Full sample. Let \(\bY = (\bY_1, \dots, \bY_N)\), \(\bX= (\bX_1', \dots, \bX_N')'\). Then \[ \small \bY = \bX\bbeta + \bU \] What are the dimensions of \(\bY_i, \bX_i, \bY, \bX\)?

Individual Matrix Form with Individual Intercepts

Model (3) in individual matrix form: \[ \bY_i = \mathbf{1}_T\alpha_i + \bU_i \] where \(\mathbf{1}_T\) — \(T\)-vector of ones

Not that insightful

Full Sample Matrix Form with Individual Intercepts

Model (3) in full sample matrix form \[ \begin{aligned} \bY & = \bF \bLambda + \bU, \\ \bLambda & = (\alpha_1, \dots, \alpha_N)', \\ \bF & = \bI_N \otimes \mathbf{1}_T, \end{aligned} \tag{4}\] where \(\otimes\) is the Kronecker product. Intuition:

• There are \(N\) regressors, \(i\)th regressor is the dummy of being the \(i\)th unit (0/1 regressor values)
• \(\bLambda\) — associated parameter vector

More Complex Example: Two-Way Intercept Model

Now consider more general model: \[ Y_{it} = \alpha_i + \gamma_t + U_{it} \] Here want to treat both \(\alpha_i\) and \(\gamma_t\) as parameters

Individual matrix form: \[ \begin{aligned} \bY_i & = \mathbf{1}_T \alpha_i + \bI_T \bgamma + \bU_i\\ \bgamma & = (\gamma_1, \dots, \gamma_T)' \end{aligned} \]

Two-Way Model: Full Sample Matrix Form

Can write \[ \begin{aligned} \bY & = \bF\bLambda + \bU, \\ \bF & = \left(\bI_N \otimes \mathbf{1}_T, \mathbf{1}_N\otimes \bI_T \right)\\ \bLambda & = (\alpha_1, \dots, \alpha_N, \gamma_1, \dots, \gamma_T) \end{aligned} \]

• Both \(\alpha_{\cdot}\) and \(\gamma_{\cdot}\) treated as parameters
• Regressors in \(\bF\): \(N\) dummy regressors from before; \(T\) new dummy regressors, \(t\)th new regressor — indicator of \(t\)th period

Adding Other Covariates

Can write Equation 2 as \[ Y_{it} = \alpha_i + \gamma_t + \bX_{it}'\bbeta + U_{it}, \] for \(\bX_{it} = (D_{it})\) and \(\bbeta = (\delta)\)

More generally, consider any vector \(\bX_{it}\) — not just binary treatments

Its matrix form is \[ \bY = \bF\bLambda + \bX\beta + \bU \]

Random-Intercept (Fixed Effects) Models

Definition 1 Models of the kind \[ \small \bY = \bF\bLambda +\bX\bbeta + \bU, \tag{5}\] where \(\bF\) is a matrix of 0s and 1s are called fixed effects or random intercept models

• Fixed effects and random intercepts — often used interchangeably
• Random intercepts — less ambiguous

Examples of Model (5)

• Individual fixed effects/intercepts (one-way) \[ Y_{it} = \alpha_i + \bX_i'\bbeta + U_{it} \]
• Two-way models (time and individual effects): \[ Y_{it} = \alpha_i + \gamma_t + \bX_i'\bbeta + U_{it} \]
• Can include more complicated effects, see empirical illustration (where \(i\) — US counties, \(t\) — quarters; effects — county-season and state-year)

Fixed Effect (Within) Estimators

Estimation Strategies

Suppose \(\E[U_{it}|\bX_i]=0\). How to estimate parameters of Model (5)?

There are two main strategies:

• Estimate including \(\bF\) and \(\bX\) as regressors — least squares dummy variable (LSDV) estimator
• Get rid of \(\bF\), estimate after — within estimator

LSDV Estimation

LSDV — simply regress \(\bY\) on \((\bF, \bX)\): \[ (\hat{\bLambda}, \hat{\bbeta}^{LSDV}) = \argmin_{\bL, \bb} \norm{\bY - \bF\bL -\bX\bb }_2^2 \]

For example with two-way effects:

\[\small \begin{aligned} & \left(\hat{\alpha}_1, \dots, \hat{\alpha}_N, \hat{\gamma}_1, \dots, \hat{\gamma}_T, \hat{\bbeta}^{LSDV} \right)\\ & = \argmin_{a_1, \dots, a_N, g_1, \dots, g_T, \bb}\sum_{i=1}^N \sum_{t=1}^T \left(Y_{it} - a_i - g_t - \bX_{it}'\bb \right)^{2} \end{aligned} \]

Within Estimation I: One-Way Transformation

First consider one-way model \(Y_{it} = \alpha_{i} + \bX_{it}' + U_{it}\). For \(\bW_{it} = Y_{it}, \bX_{it}, U_{it}\), define the (one-way) within-transformed version of \(W_{it}\) as \[ \small \tilde{W}_{it} = W_{it} - \dfrac{1}{T}\sum_{s=1}^T W_{is} \tag{6}\]

Within transformation eliminated fixed effects (=\(\bF\)) \[ \small \tilde{Y}_{it} = \tilde{\bX}_{it}'\bbeta + \tilde{U}_{it} \]

Within Estimation II: Two-Way Transformations

Suppose \(Y_{it} = \alpha_i + \gamma_t + \bX_{it}'\bbeta + U_{it}\). Define (two-way) within-transformed variables as \[ \small \tilde{W}_{it} = W_{it} - \dfrac{1}{T}\sum_{s=1}^T W_{is} - \dfrac{1}{N} \sum_{j=1}^N W_{jt} + \dfrac{1}{NT} \sum_{j=1}^N \sum_{s=1}^T W_{js} \tag{7}\]

Again eliminated fixed effects (=\(\bF\)) \[ \small \tilde{Y}_{it} = \tilde{\bX}_{it}'\bbeta + \tilde{U}_{it} \]

More General Within Transformation

Consider general model: \[ \small \bY = \bF\bLambda + \bX\bbeta + \bU \]

There exists a linear transformation that eliminates \(\bF\): \[ \small \tilde{\bY} = \tilde{\bX}\bbeta + \tilde{\bU} \]

Called the FWL or the (generalized) within transformation

Within Estimation

Within estimation: just regressing \(\tilde{\bY}\) on \(\tilde{\bX}\) with OLS: \[ \hat{\bbeta}^{W} = \argmin_{\bb} \sum_{i=1}^N \sum_{t=1}^T (\tilde{Y}_{it} - \tilde{\bX}_{it}'\bb)^2 \]

• \(\tilde{\bX}\) must have maximum column rank (no collinearity in transformed regressors)
• Intuition in one-way case (only \(\alpha_i\)): some variation in \(\bX_{it}\) over time

Equivalence of Approaches

Proposition 1 \[ \hat{\bbeta}^{LSDV} = \hat{\bbeta}^{W} \]

• Both approaches: same estimated values
• Explains why we got same results from two regression approaches to DiD last time
• Allows to use single name for both estimators. Usually called fixed effects or random intercept estimators

Which Approach to Use?

When to use LSDV vs. within estimation?

• LSDV: only when you care about \(\Lambda\) and they have some economic meaning. Example: \(\alpha_i\) is the innate skill of worker \(i\) (e.g. de la Roca and Puga 2017)
• Within: in all other cases

Sometimes impossible to compute LSDV estimator: number of fixed effects is too large to even simply store the data matrix:

• Called the “high-dimensional” fixed effect case
• In practice the within transformation is cleverly done indirectly (Correia 2016)

Pooled OLS

Another special case of model (5) — pooled OLS:

• No fixed effects, directly regressing \(Y_{it}\) on \(\bX_{it}\)
• Fully ignoring the panel structure
• Involves no transformations and no \(\bF\)
• Interpretation: the least flexible example of (5)

Implementation in Python

• linearmodels supports both LSDV and within transformations
  • Defaults to eliminating effects
  • LSDV can be used with PanelOLS.fit(use_lsdv=True)
• pyfixest was designed for high-dimensional FE estimation (can handle small examples too)
  • LSDV not available (to the best of my knowledge)
  • See empirical application for example usage

Causal Properties with General Treatment

Question: Causal Properties of FE Estimators

So far:

• Now have described the estimation approach underlying DiD regression estimation
• Noticed that it can handle general treatments \(\bX_{it}\), not just scalar binary \(D_{it}\)

What are the causal properties of such estimators? Under which models do they give meaningful results?

Reflection on our Approach

Note:

• Approach this problem differently from past lectures:
  • Here first formulate an estimator and only after start understanding causal properties
  • Usually goes the other way: fix causal problem, try to figure out identification and estimation
• Doing so for historic reasons — these estimators came first, serious causal thinking more recently

Models Considered

Will consider two kinds of models under strict exogeneity

• Random intercept causal process with same \(\bbeta\) for everyone — reflects estimator structure
• Model with heterogeneous effects \(\bbeta_i\)

Properties under Random Intercept Model

Causal Framework with Random Intercepts

• Some vector of treatments \(\bX_{it}\)
• Potential outcome of unit \(i\) in time \(t\) given by \[ Y^{\bx}_{it} = \alpha_i + \gamma_t + \bx'\bbeta + U_{it} \tag{8}\]
• Will think about about exogeneity conditions later
• Object of interest — \(\bbeta\) (plays the role of ATE, ATT, …)

For definiteness, we do two-way effects, but can apply same analysis for any configuration of random intercepts, just need to define \(\tilde{Y}_{it}\) appropriately

Sampling Setting

Work in the following setting

• Units drawn IID
• \(N\) large, \(T\) fixed
  • More typical kind of panel data (“micro” panel)
  • Contrast with “large” panel data with both \(N\) and \(T\) large

Causal Framework: Discussion of Model (8)

• Contrast with less flexible causal model discussed before: \[ Y_{it}^{\bx} = \bx'\bbeta + U_{it} \]
• Interpretations:
  • \(\alpha_i\) — often some characteristic of \(i\) that does not change over \(t\) in sample (e.g. innate intellect)
  • \(\gamma_t\) — shocks that affect everyone equally
  • Similar logic for other types of random interecepts

Estimator for \(\bbeta\)

• Estimate \(\bbeta\) with the FE estimator, expressed as (Proposition 1) \[ \small \begin{aligned} \hat{\bbeta}^{FE} & = \left(\sum_{i=1}^N \sum_{t=1}^T \tilde{\bX}_{it}\tilde{\bX}_{it}' \right)^{-1}\sum_{i=1}^N \sum_{t=1}^T \tilde{\bX}_{it}\tilde{Y}_{it} \\ & = \left( \sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i \right)\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bY}_i \end{aligned} \] with variables transformed as in Equation 7
• Will discuss nonsingularity assumptions a bit later

Probability Limit of \(\hat{\bbeta}^{FE}\)

Realized data satisfies \(\tilde{\bY} = \tilde{\bX}_{it}'\bbeta + \tilde{U}_{it}\)

So as \(N\to\infty\) and \(T\) is fixed: \[ \small \hat{\bbeta}^{FE} \xrightarrow{p} \bbeta + \left( \E[\tilde{\bX}_{i}'\tilde{\bX}_i]\right)^{-1} \E[\tilde{\bX}_{i}'\tilde{\bU}_{i}] \]

\(T\) fixed — basically treat each unit a single \(T\)-dimensional observation (as in \(\tilde{\bU}_i\))

Rank (Nonsingularity) Conditions

So far needed to impose nonsingularity conditions

• Sample: on \(\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i\)
• Population: on \(\E[\tilde{\bX}_{i}'\tilde{\bX}_i]\)

What does it require of \(\bX_{it}\)?

• No collinearity
• Also variation after the within transformation (one-way: variation over time; two-way: different variation over time for different units; etc.)

Towards Exogeneity Conditions

For consistency want \[ \small \E[\tilde{\bX}_i'\tilde{\bU}_i] = \sum_{t=1}^T \E\left[\tilde{\bX}_{it} \tilde{U}_{it} \right] =0 \]

Sufficient that for all \(t\) \[ \small \E\left[\tilde{\bX}_{it} \tilde{U}_{it} \right] =0 \tag{9}\]

What does this condition require of \(\bX_{it}\) and \(U_{it}\)?

Exogeneity in the One-Way Case

Under one-way transformation (6), Equation 9 becomes \[ \scriptsize \E\left[\tilde{\bX}_{it} \tilde{U}_{it} \right] = \E\left[ \bX_{it}U_{it} - \dfrac{\bX_{it}}{T}\sum_{s=1}^T U_{is} - \dfrac{U_{it}}{T}\sum_{r=1}^T \bX_{ir} + \dfrac{1}{T^2} \sum_{s=1}^T\sum_{r=1}^T \bX_{is} U_{ir}\right] = 0 \]

Here would be sufficient that for all \(t\), \(s\) \[ \small \E[\bX_{it}U_{is}] = 0 \tag{10}\] Intuition: \(\bX_{it}\) and \(U_{is}\) are uncorrelated across all points in time

Strict Exogeneity in the Panel Case

What about beyond one-way effects? Usually impose an assumption that covers all cases — panel data version of strict exogeneity :

Assumption (strict exogeneity): \[ \E[U_{it} | \bX_{i1}, \dots, \bX_{iT}] =0 \]

Much stronger than just \(\E[U_{it}|\bX_{it}]=0\)

Strict Exogeneity Implies Equation 9

Proposition 2 Let \(\E[U_{is} | \bX_{i1}, \dots, \bX_{iT}] =0\) for all \(s\). Then for any within transformation it holds for all \(t\) that \[ \E[\tilde{\bX}_{i}'\tilde{\bU}_i] =0 \]

• Proof by properties of conditional expectations
• Covers all configurations of random intercepts

Consistency Result

Proposition 3 Let

1. \((\bX_i, \bU_i)\) be IID and model (8) be true
2. \(\E[\tilde{\bX}_i'\tilde{\bX}_i]\) exist and be invertible
3. \(\E[U_{it} | \bX_{i1}, \dots, \bX_{iT}] =0\)

Then as \(N\to\infty\)

\[ \hat{\bbeta}^{FE} \xrightarrow{p} \bbeta \]

Asymptotic Distribution

Proposition 4 Let

1. \((\bX_i, \bU_i)\) be IID and model (8) be true
2. \(\E[\tilde{\bX}_i'\tilde{\bX}_i]\) exist and be invertible; \(\E\left[\norm{\tilde{\bX}_i'\tilde{\bU}_i}^2\right]<\infty\)
3. \(\E[U_{it} | \bX_{i1}, \dots, \bX_{iT}] =0\)

Then as \(N\to\infty\)

\[ \scriptsize \sqrt{N}\left(\hat{\bbeta}^{FE} - \bbeta\right) \xrightarrow{d} N\left(0, \left(\E[\tilde{\bX}_i'\tilde{\bX}_i] \right)^{-1} \E[\tilde{\bX}_i'\tilde{\bU}_i\tilde{\bU}_i'\tilde{\bX}_i] \left(\E[\tilde{\bX}_i'\tilde{\bX}_i] \right)^{-1}\right) \]

Discussion of Asymptotic Results for \(\hat{\bbeta}^{FE}\)

• Can do inference and estimate errors in more or less standard way (confidence intervals, hypothesis tests, …)
• Proof of asymptotic normality — exercise
• Not examinable technical point: some within transformations (e.g. two-way) can create dependence across \(i\). This dependence disappears as \(N\to\infty\) and does not affect asymptotics

Properties under Heterogeneous Coefficient Model

Causal Framework with Heterogeneous Coefficients

Consider different potential outcomes setting: \[ \small Y_{it}^{\bx} = \bx'\bbeta_{\textcolor{teal}{i}} + U_{it} \tag{11}\] under strict exogeneity \(\E[U_{it} | \bX_{i1}, \dots, \bX_{iT}] =0\)

• Special case: unit-specific intercepts
• Does not nest two-way or other random intercepts with time variation
• (11) interesting because it allows heterogeneous effects of same change in \(\bX_{it}\)

FE Estimator

Can still use the FE estimator: \[ \hat{\bbeta}^{FE} = \left(\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i \right)^{-1}\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bY}_i \] Can do any transformation such that \(\E[\tilde{\bX}_i'\tilde{\bX}_i]\) is invertible

What does \(\hat{\bbeta}^{FE}\) do under model (11)?

Expanding Estimator and Taking Limits

Substituting model (11) gets us \[ \small \begin{aligned} \hat{\bbeta}^{FE} & = \left(\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i \right)^{-1}\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i\bbeta_i + \left(\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bX}_i \right)^{-1}\sum_{i=1}^N \tilde{\bX}_i'\tilde{\bU}_i\\ & \xrightarrow{p} \E\left[ \bW(\tilde{\bX}_i) \bbeta_i\right] \end{aligned} \] for \(\small\bW(\tilde{\bX}_i) = \left(\E\left[\tilde{\bX}_i'\tilde{\bX}_i\right] \right)^{-1} \tilde{\bX}_i'\tilde{\bX}_i\)

Discussion of \(\hat{\bbeta}^{FE}\) under Heterogeneous Effects

• Result: FE estimator estimate weighted average of \(\bbeta_i\)
  • Weights are positive definite and sum to one
    
  • Weights depend on within transformation used
• In general \(\E\left[ \bW(\tilde{\bX}_i) \bbeta_i\right]\neq \E[\bbeta_i]\) (except RCTs)
• Careful: \(\E\left[ \bW(\tilde{\bX}_i) \bbeta_i\right]\) and \(\E[\bbeta_i]\) can even have different signs sometimes

Extension: Factor Models

Can generalize random intercept models to the form \[ Y_{it}^{\bx} = \balpha_i'\bgamma_t + \bx'\bbeta_i + U_{it} \]

• \(\bgamma_t\) — unobserved “factors”, \(\balpha_i\) — factor loadings
• Allows people to react differently to same shocks in \(\bgamma_t\)
• See chapter 29 in Pesaran (2015)

Empirical Application

Context and Setting

Empirical Question

Recall empirical question:

How does pollution affect labor market outcomes?

• Want to answer question without instruments
• Need some “economic activity-exogenous” measure of pollution
• Also need to control “predisposition” to such pollution

Data

Use data from Borgschulte, Molitor, and Zou (2024)

• Quarterly data for all 3142 counties in the US over 2007-2019 (\(T\approx 48\))
• Data on average earnings and employment in county

# Load data
columns_to_load = [
    "countyfip",                # FIPS
    "rfrnc_yr",                 # Year
    "rfrnc_qtroy",              # Quarter 
    "d_pc_qwi_payroll",         # Earnings (annual diff) 
    "hms_deep",                 # Number of smoke days
    "fe_countyqtroy",
    "fe_styr",
    "fe_stqtros",
    "seer_pop",                 # Population 
]
county_df = pd.read_csv(
  "data/county_quarter.tab", 
  sep="\t", 
  usecols=columns_to_load,
)

# Rename columns
column_rename_dict = {
  "countyfip":"fips",
  "rfrnc_yr":"year",
  "rfrnc_qtroy":"quarter",
  "d_pc_qwi_payroll":"diff_payroll", 
  "hms_deep":"smoke_days",
  "fe_countyqtroy":"fe_id_county_quarter",
  "fe_styr":"fe_id_state_year",
  "fe_stqtros":"fe_id_state_quarter",
  "seer_pop":"population",
}
county_df = county_df.rename(columns=column_rename_dict)
 
# View data
county_df.dropna(inplace=True)
county_df.head(2)

	fips	year	quarter	smoke_days	population	fe_id_state_year	fe_id_county_quarter	fe_id_state_quarter	diff_payroll
4	1001	2007	1	0.0	52405.0	2	1	5	25.800293
5	1001	2007	2	9.0	52405.0	2	2	6	33.243042

Pollution Measure

Pollution — number of smoke days because of wildfires

• Wildfire smoke can travel far — “exogenous”
• Some places at great risk of fires or persistent smoke — how to handle impact?

Distribution of Pollution Measure

Specification and Estimation

Key Variables and Homogeneity Assumption

• Outcome: change in average earnings (employment — exercise)
• Treatment: number of smoke days in quarter
• Analysis level: \(i\) — counties, \(t\) — quarters (no aggregation)
• Assumption: treatment has same effect in all \((i, t)\) and at all levels. Assumed potential outcomes model \[\small (\Delta \text{Earnings}_{it})^{\text{Smoke days}} = \beta\text{Smoke days} + \text{FEs} + U_{it} \]

Random Intercepts

Need to choose random intercepts/FEs so that strict exogeneity holds

Specification of Borgschulte, Molitor, and Zou (2024): include

• County-season intercepts (capture effect of geography, differing per season)
• State-year (capture overall economic trends)

About 13000 different random intercepts (a bit more complicated than \(\alpha_i\) and \(\delta_t\))

Estimation

• Will use pyfixest this time to estimate
• feols for fixed effect estimation
• Regression formula in fml, random intercepts after |

import pyfixest as pf

results = pf.feols(
    fml="diff_payroll ~ smoke_days | fe_id_state_year + fe_id_county_quarter", 
    data=county_df, 
    vcov={"CRV1": "fips + fe_id_state_quarter",}, 
    weights="population",
)

Estimation Results

• An additional day reduces quarterly earning about $5.20 on average — significant effect
• Clustered standard errors (p. 77 in Cunningham 2021)

pf.etable(results)

	diff_payroll
	(1)
coef
smoke_days	-5.217*** (0.774)
fe
fe_id_county_quarter	x
fe_id_state_year	x
stats
Observations	160346
S.E. type	by: fips+fe_id_state_quarter
R2	-
Significance levels: * p < 0.05, ** p < 0.01, *** p < 0.001. Format of coefficient cell: Coefficient (Std. Error)

Recap and Conclusions

Recap

In this lecture we

1. Discussed fixed effect estimators for DiD and beyond
   • LSDV
   • Within transformation
2. Proved causal properties of FE estimators under
   • A random intercept model
   • Model with unit-specific coefficients

Next Questions

• What if you want to estimate \(\E[\bbeta_i]\)?
• When does strict exogeneity fail? Can you relax it?
• What does panel data let you do in nonlinear settings?

References

Baltagi, Badi H. 2021. Econometric Analysis of Panel Data. Springer Texts in Business and Economics. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-53953-5.

Borgschulte, Mark, David Molitor, and Eric Yongchen Zou. 2024. “Air Pollution and the Labor Market: Evidence from Wildfire Smoke.” Review of Economics and Statistics 106 (6): 1558–75. https://doi.org/10.1162/rest_a_01243.

Correia, Sergio. 2016. “A Feasible Estimator for Linear Models with Multi-Way Fixed Effects.”

Cunningham, Scott. 2021. Causal Inference: The Mixtape. Yale University Press. https://doi.org/10.2307/j.ctv1c29t27.

de la Roca, Jorge, and Diego Puga. 2017. “Learning By Working in Big Cities.” Review of Economic Studies 84 (1): 106–42. https://doi.org/10.1093/restud/rdw031.

Hansen, Bruce. 2022. Econometrics. Princeton_University_Press.

Huntington-Klein, Nick. 2025. The Effect: An Introduction to Research Design and Causality. S.l.: Chapman and Hall/CRC.

Pesaran, M. Hashem. 2015. Time Series and Panel Data Econometrics. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198736912.001.0001.

Wooldridge, Jeffrey M. 2020. Introductory Econometrics: A Modern Approach. Seventh edition. Boston, MA: Cengage.