Visualizing Regression with the FWL Theorem in R

Downloads

Each dataset is available as a labeled Stata .dta and its source file.

⇩ Download all data (ZIP)stata_codebook.do

Run stata_codebook.do in Stata once to attach long-form per-variable notes to the .dta files.

Load directly in code

Every file loads straight from GitHub (raw URLs). Swap the file name to load any dataset.

Stata

* Stata 14+ : `use` reads an https URL directly
global BASE "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_fwlplot/data/"
use "${BASE}store_data.dta", clear
describe
notes

Python

!pip install -q pyreadstat
import pandas as pd
BASE = "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_fwlplot/data/"
df = pd.read_stata(BASE + "store_data.dta")

# load every dataset at once
files = ["store_data", "flights_sample", "wagepan"]
data = {f: pd.read_stata(BASE + f + ".dta") for f in files}

# pyreadstat (richest metadata) reads LOCAL files -> download first
import pyreadstat, urllib.request
urllib.request.urlretrieve(BASE + "store_data.dta", "store_data.dta")
df, meta = pyreadstat.read_dta("store_data.dta")

Copy and paste this snippet in Google Colab app. https://colab.research.google.com/notebooks/empty.ipynb

R

# R : haven::read_dta auto-downloads an https URL
library(haven)
BASE <- "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_fwlplot/data/"
df <- read_dta(paste0(BASE, "store_data.dta"))

Overview & sources

Companion data for a hands-on R tutorial on the fwlplot package (Butts & McDermott, 2024), which renders the Frisch–Waugh–Lovell (FWL) theorem as a picture: any multiple-regression coefficient equals the slope of a simple bivariate regression after partialling the other controls out of both axes. The post builds intuition across three datasets — an n=200 simulated retail panel where income confounds the coupon–sales relationship, the nycflights13 flights data (a 5,000-row cleaned sample), and Wooldridge's wagepan panel (545 individuals over 1980–1987). The simulated case shows confounding reverse the naive coupon slope from −0.093 to the controlled +0.212 (true effect +0.2); fixed effects on the flights and wage panels show what "controlling for" looks like geometrically.

Data sources

Cite this data

Please cite this dataset as follows.

APA

Mendez, C. (2026). Visualizing Regression with the FWL Theorem in R [Data set]. https://carlos-mendez.org/post/r_fwlplot/

Butts, K., & McDermott, G. (2024). fwlplot: Scatter Plot After Residualizing. CRAN. https://cran.r-project.org/package=fwlplot — Frisch, R., & Waugh, F. V. (1933). Partial Time Regressions as Compared with Individual Trends. Econometrica, 1(4), 387–401. — Lovell, M. C. (1963). Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis. JASA, 58(304), 993–1010. — Wickham, H. (2021). nycflights13: Flights that Departed NYC in 2013. CRAN.

BibTeX

@misc{mendez2026rfwlplot,
  author       = {Mendez, Carlos},
  title        = {Visualizing Regression with the FWL Theorem in R},
  year         = {2026},
  howpublished = {\url{https://carlos-mendez.org/post/r_fwlplot/}},
  note         = {Data set}
}

@misc{butts2024fwlplot,
  author = {Butts, Kyle and McDermott, Grant},
  title  = {fwlplot: Scatter Plot After Residualizing},
  year   = {2024}, howpublished = {CRAN}, note = {R package},
  url    = {https://cran.r-project.org/package=fwlplot}
}
@article{frisch1933partial,
  author  = {Frisch, Ragnar and Waugh, Frederick V.},
  title   = {Partial Time Regressions as Compared with Individual Trends},
  journal = {Econometrica}, volume = {1}, number = {4}, pages = {387--401}, year = {1933}
}
@article{lovell1963seasonal,
  author  = {Lovell, Michael C.},
  title   = {Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis},
  journal = {Journal of the American Statistical Association},
  volume  = {58}, number = {304}, pages = {993--1010}, year = {1963}
}

Variable explorer search & filter all 57 variables

Type to filter by name or label, or use the chips to filter by type. Each row shows a mini distribution. Click a header to sort.

Cross-file variable index

Which file each variable appears in (● = present).

Construction & formulas

The Frisch–Waugh–Lovell (FWL) theorem: in the regression Y = X₁β₁ + X₂β₂ + ε, the coefficient β₁ on the variable of interest equals the slope from a simple bivariate regression after partialling X₂ out of both axes:

• Step 1 — residualize the outcome: regress Y on the controls X₂, keep the residuals &Ytilde; = M₂ Y.
• Step 2 — residualize the regressor: regress X₁ on the controls X₂, keep the residuals X₁tilde = M₂ X₁.
• Step 3 — residual-on-residual: regress &Ytilde; on X₁tilde; the slope equals β₁ exactly (0.212288 in the store data, matching feols() to six decimals).

Here M₂ = I − X₂(X₂'X₂)⁻¹X₂' is the residual-maker matrix. Fixed effects are FWL applied to group dummies: including | origin + dest (flights) or | nr (wages) demeans each variable within group before fitting. fwl_plot() automates all of this and plots the residualized scatter (an added-variable plot) with the regression line overlaid.

Omitted variable bias: bias = γ × δ, where γ is the coefficient on the omitted control in the full model and δ is the slope from regressing the omitted control on the regressor (income on coupons, not the reverse). In the store data, 0.3004 × (−1.0174) ≈ −0.3057, which equals the naive slope (−0.0934) minus the controlled slope (+0.2123) exactly.

The datasets

Switch datasets with the tabs. Each shows the full variable dictionary plus a sortable statistics table with mini distributions and data coverage.

Expand all datasets expand to search (Ctrl/⌘+F) or print across all datasets

Known limitations & caveats

• store_data is simulated. Values come from a known data-generating process (set.seed(42)); the true coupon effect is +0.2 by construction. It illustrates confounding and FWL, not a real retail market.
• flights_sample is a 5,000-row sample drawn (set.seed(123)) from ~317,578 cleaned flights — kept only delays in (−30, 120) minutes with no missing air time/origin/destination, and routes with more than one observation. Slopes in the post's tables are estimated on the full cleaned data, not this sample; use the sample for plotting/teaching, not for headline estimates.
• wagepan covers employed men only, 1980–1987. Time-invariant traits (educ, black, hisp) drop under individual fixed effects; the linear exper term drops under two-way (individual + year) fixed effects because experience increments by one year for everyone (collinear with year dummies).
• FWL is exact only for linear regression. For logit/Poisson and other nonlinear models the residualized scatter is at best an approximation of the conditional relationship.