Frisch–Waugh–Lovell theorem

The Frisch–Waugh–Lovell theorem states that a linear-regression coefficient on selected predictors is unchanged when other predictors are first projected out of both the outcome and those selected predictors.

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Partial regression: A regression coefficient measuring the association between variables after controlling for specified covariates. The theorem gives partial regression a concrete residual-based calculation.
Ordinary least squares: A method that estimates linear-model coefficients by minimizing the sum of squared residuals. The theorem concerns coefficients from this estimation method.
Omitted-variable bias: Bias in a regression coefficient caused by excluding a relevant variable correlated with included predictors. The theorem clarifies how included controls alter a target coefficient by removing shared variation.
Partial correlation: The correlation between two variables after linear effects of specified controls are removed from each. It measures residual association, whereas the theorem equates regression coefficients.
Orthogonal projection: The mapping of a vector onto a subspace so that its residual is perpendicular to that subspace. Regressing out controls is precisely projection onto their column space.
Least squares: An estimation principle that chooses parameters to minimize squared discrepancies between observations and predictions. Ordinary least squares and the theorem's projection interpretation rest on this principle.
Fixed effects model: A model that controls for unobserved, unit-specific factors that remain constant across observations or time. Demeaning in fixed-effects estimation is a residualization operation.
Frisch–Waugh–Lovell theorem for generalized linear models: A family of proposed extensions of the Frisch–Waugh–Lovell result to nonlinear-link regression models. The exact linear-regression identity does not generally transfer unchanged to nonlinear models.
Residualization: The process of subtracting a variable's fitted values from its observed values. The theorem residualizes both the outcome and target predictors against controls.
Column space: The set of all linear combinations of a matrix's columns. Controls remove the components lying in their design matrix's column space.