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Econometrics-I-4 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Busines
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Applied Econometrics 4. Least Squares Algebra: Partial Regression and Correlation
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Frisch-Waugh (1933) Theorem Context :  Model contains two sets of variables:         X   =  [ [1, time ] | [  other variables ]]             =  [ X 1    X 2 Regression model:         y   =   X 1 β 1   +  X 2 β 2  +  ε    (population)           =   X 1 b 1   +  X 2 b 2  +  e   (sample) Problem :  Algebraic expression for the second set  of least squares coefficients,  b 2  
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Partitioned Solution Method of solution (Why did F&W care?  In 1933, matrix  computation was not trivial!) Direct manipulation of normal equations (text p. 26-27)  produces       b 2   =  ( X 2 X 2 ) -1 X 2 ( y  -  X 1 b 1 )             What is this?   Regression of ( y  -  X 1 b 1 ) on  X 2 Important result (perhaps not fundamental).  Note the result  if  X 2 X 1  =  0 .         Useful: Probably       Likely?  Don’t count on it.
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Partitioned Inverse     Use of the partitioned inverse result produces a  fundamental result: What is the southeast  element in the inverse of the moment matrix?  1 1 1 2 2 1 2 2 -1 X 'X X 'X X 'X X 'X
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Partitioned Inverse      The algebraic result is:      [ ] -1 (2,2)   =  {[ X 2 X 2 ] -  X 2 X 1 ( X 1 X 1 )-1 X 1 X 2 } -1                =  [ X 2 ’( I  -  X 1 ( X 1 X 1 ) -1 X 1 ’) X 2 ] -1                  =  [ X 2 M 1 X 2 ] -1   Note the appearance of an “ M ” matrix.  How do  we interpret this result? Note the implication for the case in which  X 1  is a  single variable.  (Theorem, p. 28) Note the implication for the case in which  X 1  is  the constant term. (p. 29)
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Frisch-Waugh Result
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