Simulated fast-food restaurants (FWL tutorial)

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⇩ 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

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Stata

* Stata 14+ : `use` reads an https URL directly
global BASE "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/python_fwl/data/"
use "${BASE}fwl_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/python_fwl/data/"
df = pd.read_stata(BASE + "fwl_store_data.dta")

# load every dataset at once
files = ["fwl_store_data", "fwl_restaurant_panel"]
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 + "fwl_store_data.dta", "fwl_store_data.dta")
df, meta = pyreadstat.read_dta("fwl_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/python_fwl/data/"
df <- read_dta(paste0(BASE, "fwl_store_data.dta"))

Overview & sources

Companion data for the tutorial The FWL Theorem: Making Multivariate Regressions Intuitive, inspired by Courthoud (2022). A fast-food chain runs 50 simulated restaurants, one per neighborhood. Each restaurant hands out 100 discount coupons on a single day and counts how many are redeemed during the month: coupons is that redemption rate in percent, and sales is the restaurant's monthly sales. The data are built so that the answer is known in advance: a higher redemption rate raises monthly sales by exactly +0.2 per percentage point, but neighborhood income lowers redemption and raises sales, so it confounds the comparison. Ignoring income, the slope of sales on coupons is −0.1059 (p = 0.365); controlling for income it becomes +0.2673 (p = 0.031), and the Frisch–Waugh–Lovell (FWL) theorem recovers the same +0.2673 from a regression of residualized sales on residualized coupons. The appendix repeats the promotion in June and applies FWL to the two-month panel (restaurant and two-way fixed effects, first differences).

Data sources

Cite this data

Please cite this dataset as follows.

APA

Mendez, C. (2026). Simulated fast-food restaurants (FWL tutorial) [Data set]. https://carlos-mendez.org/post/python_fwl/

Courthoud, M. (2022). Understanding the Frisch-Waugh-Lovell Theorem. Towards Data Science (originally published as "The FWL Theorem, Or How To Make All Regressions Intuitive"). https://towardsdatascience.com/the-fwl-theorem-or-how-to-make-all-regressions-intuitive-59f801eb3299/ — 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. Journal of the American Statistical Association, 58(304), 993–1010.

BibTeX

@misc{mendez2026pythonfwl,
  author       = {Mendez, Carlos},
  title        = {Simulated fast-food restaurants (FWL tutorial)},
  year         = {2026},
  howpublished = {\url{https://carlos-mendez.org/post/python_fwl/}},
  note         = {Data set}
}

@misc{courthoud2022fwl,
  author       = {Courthoud, Matteo},
  title        = {Understanding the {Frisch-Waugh-Lovell} Theorem},
  year         = {2022},
  howpublished = {Towards Data Science},
  note         = {Originally published as ``The {FWL} Theorem, Or How To Make All Regressions Intuitive''},
  url          = {https://towardsdatascience.com/the-fwl-theorem-or-how-to-make-all-regressions-intuitive-59f801eb3299/}
}
@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 6 variables

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Cross-file variable index

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

Construction & formulas

Data-generating process

Simulated by simulate_store_data(n=50, seed=42) in script.py with numpy's default_rng(42); N(mean, SD) denotes a normal draw.

• income ~ N(50, 10)
• dayofweek ~ Uniform{1, 2, …, 7} (rng.integers(1, 8))
• coupons = 60 − 0.5 × income + u,  u ~ N(0, 5)
• sales = 10 + 0.2 × coupons + 0.3 × income + 0.5 × dayofweek + e,  e ~ N(0, 3)

The true causal effect of coupons on sales is +0.2 by construction: one percentage point of redemption (one more of the 100 coupons redeemed) adds 0.2 thousand US dollars (200 US dollars) of monthly sales. sales is computed from the unrounded draws; sales, coupons and income are then rounded to 2 decimals.

The June round (panel appendix)

simulate_restaurant_panel(n=50, seed=42) replays the January draws exactly and continues the same random stream. January's sales noise α ~ N(0, 3) is a persistent restaurant trait that raises sales in both months; in June it also raises redemption.

• incomeJun = incomeJan + N(0, 2); dayofweekJun ~ Uniform{1, …, 7}
• couponsJun = 60 − 0.5 × incomeJun + α + N(0, 4)
• salesJun = 13 + 0.2 × couponsJun + 0.3 × incomeJun + 0.5 × dayofweekJun + α + N(0, 1.5) (a summer lift of +3)

Pooled OLS with income and a June dummy gives 0.4496 (biased by the trait); restaurant fixed effects give 0.2138; two-way fixed effects and first differences both give 0.1394 (clustered SE 0.0691), which is within sampling noise of the true 0.2.

The Frisch–Waugh–Lovell (FWL) theorem

In Y = X1β1 + X2β2 + ε, let M2 = I − X2(X2′X2)−1X2′ be the residual maker for the controls (here X2 = a constant and income; M2 is symmetric and idempotent). Then

β̂1 = (X1′M2Y) / (X1′M2X1) — the multivariate coefficient equals the slope of a regression of residualized Y on residualized X1.

• Full regression sales ~ coupons + income: coupons 0.2673 (SE 0.1203, p = 0.031); income γ̂ = 0.3836.
• Residualize coupons only: coupons_tilde = residuals of coupons ~ income (uncorrelated with income, not independent of it). Regressing raw sales on coupons_tilde with no intercept returns the same 0.2673, but its SE of 1.2715 (p = 0.834) is inflated mainly by the dropped intercept: with an intercept the SE is 0.1437.
• Residualize both: sales_tilde = residuals of sales ~ income; sales_tilde ~ coupons_tilde - 1 returns 0.2673 with SE 0.1178 = 0.1203 × √(47/49). The residuals are identical to the full model's; the full model spends two extra degrees of freedom (intercept + income).
• By hand: Cov(sales_tilde, coupons_tilde) / Var(coupons_tilde) = 3.9380 / 14.7320 = 0.2673.
• Two controls (income + dayofweek partialled out together): full OLS and FWL both give 0.2706.

Omitted-variable-bias identity

Exact in the sample: β̂naive = β̂full + γ̂ × δ̂, where γ̂ is income's coefficient in the full regression and δ̂ is the slope from regressing the omitted variable (income) on coupons:

−0.1059 = 0.2673 + 0.3836 × (−0.9730) = 0.2673 − 0.3732

The auxiliary regression must run in this direction (income ~ coupons); regressing coupons on income instead (slope −0.3935) gives a product of −0.151 that reconciles nothing. In the population δ = Cov(income, coupons) / Var(coupons) = −50 / 50 = −1.0, so the naive slope converges to 0.2 + 0.3 × (−1.0) = −0.10 — confounding flips the sign of a positive effect.

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Known limitations & caveats

• Simulated, not observed. The 50 restaurants come from a known data-generating process with a true coupon effect of +0.2 by construction. The files illustrate confounding and the FWL theorem, not a real fast-food market. coupons keeps two decimals, so read 36.93 as about 37 of the 100 coupons.
• Generated in Python with numpy's PCG64 generator (np.random.default_rng(42) in script.py). sales, coupons and income are rounded to 2 decimals, and every estimate in the post is computed on these rounded values.
• Load this file to reproduce the post in R or Stata. Native simulations with seed 42 (set.seed(42) in R, set seed 42 in Stata) use different random-number generators, so they produce different draws and different estimates. Read fwl_store_data.csv or fwl_store_data.dta instead of re-simulating.
• Small sample. With n = 50 the controlled estimate (+0.2673, SE 0.1203) is close to, but not equal to, the true +0.2; the naive slope (−0.1059) sits near its population limit of −0.10.
• dayofweek is a bare 1–7 index. The simulation attaches no weekday names to the codes, and the variable enters the data-generating process linearly (slope +0.5). It is drawn independently of income and coupons (sample correlation with coupons −0.076).
• No identifier in the January file. fwl_store_data rows are in the order the restaurants were drawn, which is the order of restaurant_id 1–50 in fwl_restaurant_panel.