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# C9D1 - SIMULTANEOUS EQUATIONS BIAS p = 1 2 w u p w = 1 2 p...

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SIMULTANEOUS EQUATIONS BIAS 1 This sequence shows why OLS is likely to yield inconsistent estimates in models composed of two or more simultaneous relationships. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β

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SIMULTANEOUS EQUATIONS BIAS 2 In this example we suppose that we have data on p , the annual rate of price inflation, w , the annual rate of wage inflation, and U , the rate of unemployment, for a sample of countries. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β
SIMULTANEOUS EQUATIONS BIAS 3 We hypothesize that increases in wages lead to increases in prices and so p is positively influenced by w . w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β

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SIMULTANEOUS EQUATIONS BIAS 4 We also suppose that workers try to protect their real wages by demanding increases in wages as prices rise, but their ability to so is the weaker, the greater is the rate of unemployment ( α 2 > 0, α 3 < 0) . w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β
SIMULTANEOUS EQUATIONS BIAS 5 The model involves some circularity, in that w is a determinant of p, and p is a determinant of w . Variables whose values are determined interactively within the model are described as endogenous variables. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β p and w are endogenous U is exogenous

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SIMULTANEOUS EQUATIONS BIAS 6 We will cut through the circularity by expressing p and w in terms of their ultimate determinants, U and the disturbance terms u p and u w . Variables such as U whose values are determined outside the model are described as exogenous variables. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β p and w are endogenous U is exogenous
SIMULTANEOUS EQUATIONS BIAS p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β 7 We will start with p . The first step is to substitute for w from the second equation. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β

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SIMULTANEOUS EQUATIONS BIAS 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 + + + + = - p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β 8 We bring the terms involving p together on the left side of the equation and thus express p in terms of U , u p , and u w . w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β
SIMULTANEOUS EQUATIONS BIAS 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 p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β 9 Next we take the equation for w and substitute for p from the first equation. w u U p w + + + = 3 2 1 α α α p u w p + + = 2 1 β β

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SIMULTANEOUS EQUATIONS BIAS 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 10 p w u u U p p + + + + + = ) ( 3 2 1 2 1 α α α β β 2 2 2 3 1 2 1 1 β α α α β α α - + + + + = w p u u U w We bring the terms involving w together on the left side of the equation and thus express w in terms of U , u p , and u w .
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C9D1 - SIMULTANEOUS EQUATIONS BIAS p = 1 2 w u p w = 1 2 p...

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