Econometrics I, Frischwaugh | University of Houston

University:
University of Houston
Course:
FINA 4360 | International Financial Management
Academic year:

2022
Views:

311

Pages:

1
Author:

nadelnnl7v

ECONOMETRICS I, SPRING 2018. A “projection based” proof of the Frisch-Waugh theorem. Consider regression Y = X1 β1 + X2 β2 + e. (1) We will use three usefull facts: 1. The best fit to the least squares problem is unique (except, of course, if there is perfect collinarity). 2. Any vector or matrix of variables can be split into its projections. In particular X2 = P1 X2 + M1 X2 . This is something you can ALWAYS do. Here P1 and M1 are the projection and residual makers for X1 . (This is not so surprising because M1 = I − P1 .) 3. A regression on orthogonal (sets of) regressors can be done on each (set) at a time while still getting the coefficients from the joint regression. Now Ŷ = X1 β̂1 + X2 β̂2 , (2) Ŷ = X1 β̂1 + (P1 X2 + M1 X2 )β̂2 , (3) and, using fact 2, we have or, because P1 = X1 (X10 X1 )−1 X10 , we have Ŷ = X1 γ̂1 + (M1 X2 )β̂2 , (4) where γ̂1 = (β̂1 + (X10 X1 )−1 X10 X2 β̂2 ). Look at this expression. The fit is the same, so we are still looking at the OLS fit. The fitted coefficients are γ̂1 (the formula for which we don’t often make use of) and β̂2 (the original coefficient) to M1 X2 . Now, use fact 3. Because—by construction—M1 X2 and X2 are orthogonal, we have that the regression Y = (M1 X2 )β2 + u (5) delivers the same β̂2 as does (5) which is the same β̂2 that comes from the full equaiton (1). Finally, notice that if you instead regress (M1 Y ) = (M1 X2 )β2 + u , (6) you also get the same β̂2 . This comes from fact 2, because Y = M1 Y + P1 Y and P1 Y is orthogonal to M1 X2 , and will drop out in the regression. You may want to use the form (d) to get the same residuals as in (a).

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