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C9D4 - TWO-STAGE LEAST SQUARES p = 1 2 w u p w = 1 2 p 3U 4...

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TWO-STAGE LEAST SQUARES w u x U p w + + + + = 4 3 2 1 α α α α 1 In a second variation, we shall consider the model shown above. x is the rate of growth of productivity, assumed to be exogenous. w is now hypothesized to be a function of p , U , and x . For simplicity, p has reverted to being a simple function of w . p u w p + + = 2 1 β β
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w u x U p w + + + + = 4 3 2 1 α α α α 2 2 2 2 4 2 3 2 1 1 1 β α β β α β α β α β - + + + + + = w p u u x U p 2 2 2 4 3 1 2 1 1 β α α α α β α α - + + + + + = w p u u x U w p u w p + + = 2 1 β β 2 The reduced form equations are now as shown. The equation for w is underidentified because there is no exogenous variable to act as an instrument for p . Both U and x are already in the equation in their own right and are therefore unavailable. TWO-STAGE LEAST SQUARES
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w u x U p w + + + + = 4 3 2 1 α α α α 2 2 2 2 4 2 3 2 1 1 1 β α β β α β α β α β - + + + + + = w p u u x U p 2 2 2 4 3 1 2 1 1 β α α α α β α α - + + + + + = w p u u x U w p u w p + + = 2 1 β β 3 However, the equation for p is now overidentified. An equation is said to be overidentified if the number of available instruments exceeds the number of endogenous explanatory variables used as explanatory variables. TWO-STAGE LEAST SQUARES
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w u x U p w + + + + = 4 3 2 1 α α α α 2 2 2 2 4 2 3 2 1 1 1 β α β β α β α β α β - + + + + + = w p u u x U p 2 2 2 4 3 1 2 1 1 β α α α α β α α - + + + + + = w p u u x U w p u w p + + = 2 1 β β 4 We could use U as an instrument. TWO-STAGE LEAST SQUARES
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w u x U p w + + + + = 4 3 2 1 α α α α 2 2 2 2 4 2 3 2 1 1 1 β α β β α β α β α β - + + + + + = w p u u x U p 2 2 2 4 3 1 2 1 1 β α α α α β α α - + + + + + = w p u u x U w p u w p + + = 2 1 β β 5 Equally, we could use x as an instrument. Both estimators are consistent. Hence, in large samples they will yield the same estimate - the true value. But in finite samples they will be different because they have different error components.
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