What Does TWFE Actually Do? Manual Demeaning and the FWL Theorem

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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

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_demeaning_twfe/data/"
use "${BASE}source_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_demeaning_twfe/data/"
df = pd.read_stata(BASE + "source_data.dta")

# load every dataset at once
files = ["source_data", "data_demeaned"]
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 + "source_data.dta", "source_data.dta")
df, meta = pyreadstat.read_dta("source_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_demeaning_twfe/data/"
df <- read_dta(paste0(BASE, "source_data.dta"))

Overview & sources

Companion data for a hands-on R tutorial that takes the two-way fixed effects (TWFE) estimator apart to show it is nothing more than ordinary least squares applied to two-way demeaned data — the equivalence guaranteed by the Frisch–Waugh–Lovell (FWL) theorem. The tutorial uses a balanced, synthetic Barro convergence panel of 150 countries observed over 8 time periods (1,200 observations), regressing GDP-per-capita growth on log initial income, investment share, population growth, human capital, and government consumption. It estimates the model with country and time fixed effects via fixest::feols(), then replicates the coefficients by hand — subtracting country means, subtracting time means, and adding back the grand mean before running base R's lm(). The two routes match to at least 12 significant digits (the convergence coefficient is −0.055286 either way; the largest coefficient difference is 3.05×10⁻¹⁶, on the order of machine epsilon), while naive lm() standard errors understate uncertainty by 7–22% because they ignore the degrees of freedom consumed by the absorbed fixed effects.

Data sources

Cite this data

Please cite this dataset as follows.

APA

Mendez, C. (2026). What Does TWFE Actually Do? Manual Demeaning and the FWL Theorem [Data set]. https://carlos-mendez.org/post/r_demeaning_twfe/

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{mendez2026rdemeaningtwfe,
  author       = {Mendez, Carlos},
  title        = {What Does TWFE Actually Do? Manual Demeaning and the FWL Theorem},
  year         = {2026},
  howpublished = {\url{https://carlos-mendez.org/post/r_demeaning_twfe/}},
  note         = {Data set}
}

@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 17 variables

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Cross-file variable index

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

Construction & formulas

The model is a two-way fixed-effects growth regression over country i and period t:

• TWFE specification: growth_it = α_i + λ_t + β·x_it + u_it — country fixed effects α_i and time fixed effects λ_t absorb every time-invariant country trait and every country-invariant time shock.
• Two-way within (demeaning) transformation: x̃_it = x_it − x̄_i. − x̄_.t + x̄_.. — observed value minus the country mean (x̄_i., average over periods), minus the time mean (x̄_.t, average over countries), plus the grand mean (x̄_..). The _dm columns in data_demeaned are exactly this transform of the six model variables.
• Grand-mean correction: the + x̄_.. term cancels the double-subtraction of the overall level (the grand mean is removed once via x̄_i. and once via x̄_.t); without it the demeaned variables are not centered at zero.
• FWL equivalence: β̂_TWFE = β̂_OLS on demeaned data — running plain OLS (lm()) on the _dm columns returns exactly the feols() TWFE slopes (up to machine precision).

Regressor construction from the raw inputs: log_s_k = log(s_k) (log investment share), log_n_gd = log(n_pop + 0.05) (log of population growth plus the standard 0.05 for growth + depreciation), and log_hcap = log(hcap) (log human capital).

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Known limitations & caveats

• Synthetic data. The panel is simulated; coefficient values reflect the data-generating process, not real-world economic dynamics — they are not empirical evidence about growth convergence.
• Coefficients match, standard errors do not. OLS on the demeaned data reproduces the TWFE point estimates exactly, but naive lm() standard errors are too small (by 7–22% here) because they do not subtract the degrees of freedom consumed by the 150 country plus 8 time effects. Always use a dedicated panel estimator (e.g. feols()) for inference.
• Balanced panel only. The closed-form two-way demeaning used here is exact because the panel is balanced; unbalanced panels require the iterative projection that fixest performs internally.
• Demeaning kills time-invariant regressors. Any variable constant within a country over time equals its own country mean, so its demeaned column is zero everywhere and it drops out — this is why fixed-effects models cannot identify time-invariant characteristics.