en>Gbint: Added small introduction to half-\Theta graphs

2013-04-09T19:58:01Z

Added small introduction to half-\Theta graphs

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

The Frisch–Waugh–Lovell theorem states that if the [[linear regression|regression]] we are concerned with is:

:<math> Y = X_1 \beta_1 + X_2 \beta_2 + u \! </math>

where <math> X_1 </math> and <math> X_2 </math> are <math> n \times k_1 </math> and <math> n \times k_2 </math> respectively and where <math> \beta_1 </math> and <math> \beta_2 </math> are [[conformable matrix|conformable]], then the estimate of <math> \beta_2 </math> will be the same as the estimate of it from a modified regression of the form:

:<math> M_{X_1} Y = M_{X_1} X_2 \beta_2 + M_{X_1} u \!, </math>

where <math> M_{X_1} </math> projects onto the [[orthogonal complement]] of the image of the [[projection matrix]] <math> X_1(X_1'X_1)^{-1}X_1' </math>. Equivalently, ''M''''X''1 projects onto the [[orthogonal complement]] of the column space of ''X''1. Specifically,

:<math> M_{X_1} = I - X_1(X_1'X_1)^{-1}X_1'. \! </math>

This result implies that all these secondary regressions are unnecessary: 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.

==References==
*{{cite journal |first=Ragnar |last=Frisch |first2=Frederick V. |last2=Waugh |title=Partial Time Regressions as Compared with Individual Trends |journal=[[Econometrica]] |volume=1 |issue=4 |year=1933 |pages=387–401 |jstor=1907330 }}
*{{cite journal |last=Lovell |first=M. |year=1963 |title=Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis |journal=[[Journal of the American Statistical Association]] |volume=58 |issue=304 |pages=993–1010 |doi=10.1080/01621459.1963.10480682 }}
*{{cite journal |last=Mitchell |first=Douglas W. |title=Invariance of results under a common orthogonalization |journal=Journal of Economics and Business |volume=43 |issue=2 |year=1991 |pages=193–196 |doi=10.1016/0148-6195(91)90018-R }}
*{{cite journal |last=Lovell |first=M. |year=2008 |title=A Simple Proof of the FWL Theorem |journal=[[Journal of Economic Education]] |volume=39 |issue=1 |pages=88–91 |doi=10.3200/JECE.39.1.88-91 }}

{{DEFAULTSORT:Frisch-Waugh-Lovell theorem}}
[[Category:Econometrics]]
[[Category:Economics theorems]]
[[Category:Regression analysis]]
[[Category:Statistical theorems]]

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en>Gbint: Added small introduction to half-\Theta graphs