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Multiple Regression & Inferenceintermediate

OLS Estimation and Partialling Out

OLS chooses the β^j\hat{\beta}_j that minimize the sum of squared residuals ∑u^i2\sum \hat{u}_i^2 in the multiple regression. The partialling-out (Frisch-Waugh-Lovell) result shows that β^j\hat{\beta}_j equals the slope from regressing yy on the part of xjx_j left over after the other regressors are removed. So each slope already controls for everything else in the model.

Why it matters

To find the clean effect of xjx_j, first strip out the influence of the other variables on xjx_j, keep what is unexplained, then relate yy to that residual. OLS does this for every coefficient at once. That residualizing is exactly what makes a slope a partial effect.

Formulas

OLS objective
min⁡β^0,…,β^k ∑i=1n(yi−β^0−β^1xi1−⋯−β^kxik)2\min_{\hat{\beta}_0,\dots,\hat{\beta}_k}\ \sum_{i=1}^{n}\bigl(y_i-\hat{\beta}_0-\hat{\beta}_1 x_{i1}-\dots-\hat{\beta}_k x_{ik}\bigr)^2
Minimizing squared residuals gives (k+1)(k+1) first-order conditions.
Partialling-out formula
β^j=∑ir^ij yi∑ir^ij 2\hat{\beta}_j=\frac{\sum_i \hat{r}_{ij}\,y_i}{\sum_i \hat{r}_{ij}^{\,2}}
r^ij\hat{r}_{ij} is the residual from regressing xjx_j on all other regressors.

Worked examples

Scenario

You want the effect of educ on lwage net of exper using FWL by hand.

Solution

Run `regress educ exper` and `predict r_educ, resid` to get the part of schooling unrelated to experience, then `regress lwage r_educ`. The slope on r_educ matches the educ coefficient from `regress lwage educ exper`.

NoteThis is why people say a multiple-regression slope is already adjusted for the other controls.

Common mistakes

  • ✗Thinking OLS estimates each β^j\hat{\beta}_j from a separate one-variable regression. It solves all coefficients jointly through the normal equations.
  • ✗Believing partialling out residualizes yy rather than xjx_j. The denominator uses the residual variation in xjx_j that remains after the other regressors are removed.
  • ✗Assuming you can drop a regressor with a small coefficient without affecting the others. Removing it changes the residualized variation behind the remaining slopes.
  • ✗Confusing the OLS residual u^i\hat{u}_i with the unobservable error uiu_i. The residual is the fitted-model leftover, not the population error.

Revision bullets

  • •OLS minimizes ∑u^i2\sum \hat{u}_i^2 and solves all β^j\hat{\beta}_j jointly.
  • •FWL: β^j\hat{\beta}_j comes from yy regressed on xjx_j residualized against the other regressors.
  • •Partialling out is what makes each slope a ceteris paribus effect.
  • •If xjx_j has little independent variation, ∑r^ij2\sum \hat{r}_{ij}^2 is small and the slope is imprecise.

Quick check

The partialling-out interpretation says β^j\hat{\beta}_j is obtained by regressing yy on what?

Why can a multiple-regression slope be called a partial effect?

Connected topics

OLS DerivationFitted / ResidOLS VarianceMLR modelOmitted var biasMulticollin.

Sources

  1. Wooldridge, Introductory Econometrics, Ch. 3
    Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 3.2
  2. Frisch & Waugh (1933)
    Frisch & Waugh (1933), Econometrica 1(4), 387-401
How to cite this page
Dr. Phil's Quant Lab. (2026). OLS Estimation and Partialling Out. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-ols-partialling
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