FWL Theorem

Frisch-Waugh-Lovell theorem

Estimating a Subset of Regression Coefficients

1. Model Setup

Consider the linear regression model:

[ \begin{aligned} Y &= X B + U
&= [X_1 \quad X_2] \begin{bmatrix} \beta_1 \ \beta_2 \end{bmatrix} + U
&= X_1 \beta_1 + X_2 \beta_2 + U \end{aligned} ]

where:

• ( Y ) is an ( n \times 1 ) vector of observations.
• ( X = [X_1 \; X_2] ) is an ( n \times k ) matrix of regressors.
• ( \beta = (\beta_1’, \beta_2’)’ ) is the parameter vector.
• ( U ) is the disturbance term.
• ( X_1 ) contains control variables.
• ( X_2 ) contains variables of interest.

Our objective is to estimate only ( \beta_2 ).

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2. Idea: Partialling Out ( X_1 )

To isolate the effect of ( X_2 ), we remove (project out) the variation explained by ( X_1 ).

Let:

[ M_{X_1} = I - X_1 (X_1’X_1)^{-1} X_1’ ]

be the orthogonal projection matrix onto the orthogonal complement of the column space of ( X_1 ).

Key property:

[ M_{X_1} X_1 = 0 ]

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3. Projecting the Model

Pre-multiply the model by ( M_{X_1} ):

[ \begin{aligned} M_{X_1} Y &= M_{X_1} X_1 \beta_1

• M_{X_1} X_2 \beta_2
• M_{X_1} U
  &= \underbrace{M_{X_1} X_1 \beta_1}_{=0}
• M_{X_1} X_2 \beta_2
• M_{X_1} U \end{aligned} ]

Thus,

[ M_{X_1} Y = M_{X_1} X_2 \beta_2 + M_{X_1} U ]

After projection, the effect of ( X_1 ) disappears.

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4. Estimation of ( \beta_2 )

Now run OLS of ( M_{X_1} Y ) on ( M_{X_1} X_2 ):

[ \hat{\beta}2 = \left[ (M{X_1} X_2)’ (M_{X_1} X_2) \right]^{-1} (M_{X_1} X_2)’ M_{X_1} Y ]

Using properties of projection matrices:

• ( M_{X_1}’ = M_{X_1} ) (symmetric)
• ( M_{X_1}^2 = M_{X_1} ) (idempotent)

So,

[ \begin{aligned} \hat{\beta}2 &= \left[ X_2’ M{X_1}’ M_{X_1} X_2 \right]^{-1} X_2’ M_{X_1}’ M_{X_1} Y
&= \left[ X_2’ M_{X_1} X_2 \right]^{-1} X_2’ M_{X_1} Y \end{aligned} ]

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5. Interpretation

This result shows that:

• Estimating ( \beta_2 ) in the full regression of ( Y ) on ( X_1 ) and ( X_2 )
• is equivalent to:
  1. Regressing ( Y ) on ( X_1 ) and keeping residuals,
  2. Regressing each column of ( X_2 ) on ( X_1 ) and keeping residuals,
  3. Running OLS of residualized ( Y ) on residualized ( X_2 ).

This is the Frisch–Waugh–Lovell (FWL) Theorem.

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6. Key Takeaway

To estimate a subset of coefficients:

[ \boxed{ \hat{\beta}2 = (X_2’ M{X_1} X_2)^{-1} X_2’ M_{X_1} Y } ]

You can obtain ( \hat{\beta}_2 ) by partialling out the effect of ( X_1 ) before running OLS.

References

• Econometrics: FWL Theorem Algebraic Intuition (https://www.youtube.com/watch?v=bvw-eJP3sBg)
• On Cross-Fitting with Plug-in Estimators (https://www.syadlowsky.com/blog/semiparametric/2022/10/24/on-cross-fitting-with-plug-in-estimators.html)
• Double Machine Learning (https://www.causalmlbook.com/double-machine-learning.html)
• Double Machine Learning for Causal Inference: A Practical Guide (https://medium.com/@med.hmamouch99/double-machine-learning-for-causal-inference-a-practical-guide-5d85b77aa586)
• The FWL Theorem, or How to Make All Regressions Intuitive (https://towardsdatascience.com/the-fwl-theorem-or-how-to-make-all-regressions-intuitive-59f801eb3299)