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# C9D2 - w u U p w = 3 2 1 α α α 2 2 2 2 3 2 1 1 1 β α...

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Unformatted text preview: w u U p w + + + = 3 2 1 α α α 2 2 2 2 3 2 1 1 1 β α β β α β α β- + + + + = w p u u U p w p u u U p 2 2 3 2 1 1 2 2 ) 1 ( β β α β α β β α + + + + =- w p u U u w w + + + + + = 3 2 1 2 1 ) ( α β β α α w p u u U w + + + + =- 2 3 1 2 1 2 2 ) 1 ( α α β α α β α p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β p u w p + + = 2 1 β β 2 2 2 3 1 2 1 1 β α α α β α α- + + + + = w p u u U w 1 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS In the previous sequence it was asserted that the reduced form equations have two important roles. One is that they reveal violations of Assumption B.7, that the disturbance term be distributed independently of the explanatory variable(s). w u U p w + + + = 3 2 1 α α α 2 2 2 2 3 2 1 1 1 β α β β α β α β- + + + + = w p u u U p w p u u U p 2 2 3 2 1 1 2 2 ) 1 ( β β α β α β β α + + + + =- w p u U u w w + + + + + = 3 2 1 2 1 ) ( α β β α α w p u u U w + + + + =- 2 3 1 2 1 2 2 ) 1 ( α α β α α β α p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β p u w p + + = 2 1 β β 2 2 2 3 1 2 1 1 β α α α β α α- + + + + = w p u u U w INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 2 Here the reduced form equation for w reveals that u p is a determinant of it, so we would obtain inconsistent estimates if we used OLS to fit the structural equation for p . We would have a parallel problem if we used OLS to fit the structural equation for w . w u U p w + + + = 3 2 1 α α α 2 2 2 2 3 2 1 1 1 β α β β α β α β- + + + + = w p u u U p w p u u U p 2 2 3 2 1 1 2 2 ) 1 ( β β α β α β β α + + + + =- w p u U u w w + + + + + = 3 2 1 2 1 ) ( α β β α α w p u u U w + + + + =- 2 3 1 2 1 2 2 ) 1 ( α α β α α β α p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β p u w p + + = 2 1 β β 2 2 2 3 1 2 1 1 β α α α β α α- + + + + = w p u u U w INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 3 However the reduced form equation for w also provides a solution to the problem. U is a determinant of w , and by virtue of being exogenous, it is distributed independently of u p . Further, it is not an explanatory variable in its own right. 4 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑---- + =--- +-- =-- + +- + +- =---- = w w U U u u U U w w U U u u w w U U w w U U u w u w U U w w U U p p U U b i i p pi i i i p pi i i i i p pi i i i i i i 2 2 2 1 2 1 IV 2 ] [ ] [ ] [ β β β β β β w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β Thus it satisfies the three requirements for acting as an instrument for w and we will obtain a consistent estimate of β 2 if we use the IV estimator shown. 5 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑---- + =--- +-- =-- + +- + +- =----...
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C9D2 - w u U p w = 3 2 1 α α α 2 2 2 2 3 2 1 1 1 β α...

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