Frisch–Waugh–Lovell theorem

In econometrics, the Frisch–Waugh–Lovell (FWL) theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell.[1][2][3]

The Frisch–Waugh–Lovell theorem states that if the regression we are concerned with is expressed in terms of two separate sets of predictor variables:

${\displaystyle Y=X_{1}\beta _{1}+X_{2}\beta _{2}+u}$

where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are matrices, ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ are vectors (and ${\displaystyle u}$ is the error term), then the estimate of ${\displaystyle \beta _{2}}$ will be the same as the estimate of it from a modified regression of the form:

${\displaystyle M_{X_{1}}Y=M_{X_{1}}X_{2}\beta _{2}+M_{X_{1}}u,}$

where ${\displaystyle M_{X_{1}}}$ projects onto the orthogonal complement of the image of the projection matrix ${\displaystyle X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}}$. Equivalently, M_X1 projects onto the orthogonal complement of the column space of X₁. Specifically,

${\displaystyle M_{X_{1}}=I-X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}},}$

and this particular orthogonal projection matrix is known as the residual maker matrix or annihilator matrix.[4][5]

The vector ${\textstyle M_{X_{1}}Y}$ is the vector of residuals from regression of ${\textstyle Y}$ on the columns of ${\textstyle X_{1}}$.

The most relevant consequence of the theorem is that the parameters in ${\textstyle \beta _{2}}$ do not apply to ${\textstyle X_{2}}$ but to ${\textstyle M_{X_{1}}X_{2}}$, that is: the part of ${\textstyle X_{2}}$ uncorrelated with ${\textstyle X_{1}}$. This is the basis for understanding the contribution of each single variable to a multivariate regression (see, for instance, Ch. 13 in [6]).

The theorem also implies that the secondary regression used for obtaining ${\displaystyle M_{X_{1}}}$ is unnecessary when the predictor variables are uncorrelated: using projection matrices to make the explanatory variables orthogonal to each other will lead to the same results as running the regression with all non-orthogonal explanators included.

Moreover, the standard errors from the partial regression equal those from the full regression.[7]