Coopeajj

I am a graduate student in economics at a Canadian university.

FWL

The Frish-Waugh-Lovell Theorem states that if the regression we are concerned with is:

$\textstyle \mathbf {y} =\mathbf {X_{1}} \mathbf {\beta _{1}} +\mathbf {X_{2}} \mathbf {\beta _{2}} +\mathbf {u} \!$

Where $\scriptstyle \mathbf {X_{1}}$ and $\scriptstyle \mathbf {X_{2}}$ are $\scriptstyle n\times k_{1}$ and $\scriptstyle n\times k_{2}$ respectively and where $\scriptstyle \mathbf {\beta _{1}}$ and $\scriptstyle \mathbf {\beta _{2}}$ are conformable.

Then, the estimate of $\scriptstyle \mathbf {\beta _{2}}$ will be the same as the estimate from a modified regression of the form:

$\textstyle \mathbf {M_{1}} \mathbf {y} =\mathbf {M_{1}} \mathbf {X_{2}} \mathbf {\beta _{2}} +\mathbf {M_{1}} \mathbf {u} \!$

Where, $\scriptstyle \mathbf {M_{1}}$ projects onto the orthogonal compliment of $\scriptstyle \mathbf {X_{1}}$. Specifically,

$\textstyle \mathbf {M_{1}} =\mathbf {{X_{1}}({{{X_{1}}^{T}}{X_{1}}})^{-1}{X_{1}}^{T}}$

This result is extremely useful as it implies that all these secondary regressions are not necessary (i.e., using projection matrices to make the variables orthogonal to each other will lead one to the exact same results as just running the regression with all non-orthogonal included).

If we are only concerned with $\scriptstyle \beta _{2}$ we must make matrices $\scriptstyle \mathbf {X_{1}}$ and $\scriptstyle \mathbf {X_{2}}$ orthogonal. I.e., we can use orthogonal projection matricies to anihilate $\scriptstyle \mathbf {X_{1}}$.

Define:

$\textstyle \mathbf {P_{X}} =\mathbf {X(X^{T}X)^{-1}X^{T}}$

$\textstyle \mathbf {{I_{n}}-{P_{X}}={M_{X}}}$

Clearly, $\scriptstyle \mathbf {{M_{X}}X=0}$ (i.e., $\scriptstyle \mathbf {M_{X}}$ projects onto the orthogonal compliment of $\scriptstyle \mathbf {X}$.)

To estimate the parameters using OLS they