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C8D5 - INSTRUMENTAL VARIABLES Y = 1 2 X u Suppose that you...

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INSTRUMENTAL VARIABLES 1 Suppose that you have a model in which Y is determined by X but you have reason to believe that Assumption B.7 is invalid and u is not distributed independently of X . An OLS regression would then yield inconsistent estimates. u X Y + + = 2 1 β β
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INSTRUMENTAL VARIABLES However, suppose that you have reason to believe that another variable Z is related to X but is unrelated to u . We will see that we can use it to obtain consistent estimates of the parameters. As a first step, suppose that we use it as a proxy for X . 2 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - + - - - = - - + - - = - - - = 2 2 2 2 2 2 ? 2 ] [ Z Z u u Z Z Z Z X X Z Z Z Z u u X X Z Z Z Z Y Y Z Z b i i i i i i i i i i i i i β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 ( 29 - - - = 2 2 ? 2 Z Z X X Z Z b E i i i β
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INSTRUMENTAL VARIABLES 3 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - + - - - = - - + - - = - - - = 2 2 2 2 2 2 ? 2 ] [ Z Z u u Z Z Z Z X X Z Z Z Z u u X X Z Z Z Z Y Y Z Z b i i i i i i i i i i i i i β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 ( 29 - - - = 2 2 ? 2 Z Z X X Z Z b E i i i β We will demonstrate that the resulting estimates will be biased. However, we will be able to do something about the bias. To investigate the properties of b 2 ? , we first substitute for Y from the true model.
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INSTRUMENTAL VARIABLES 4 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - + - - - = - - + - - = - - - = 2 2 2 2 2 2 ? 2 ] [ Z Z u u Z Z Z Z X X Z Z Z Z u u X X Z Z Z Z Y Y Z Z b i i i i i i i i i i i i i β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 ( 29 - - - = 2 2 ? 2 Z Z X X Z Z b E i i i β We decompose the expression.
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INSTRUMENTAL VARIABLES 5 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - + - - - = - - + - - = - - - = 2 2 2 2 2 2 ? 2 ] [ Z Z u u Z Z Z Z X X Z Z Z Z u u X X Z Z Z Z Y Y Z Z b i i i i i i i i i i i i i β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 ( 29 - - - = 2 2 ? 2 Z Z X X Z Z b E i i i β Under the assumption that u is distributed independently of Z , the second term disappears when we take expectations. We see that b 2 ? is nevertheless a biased estimator of β 2 .
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INSTRUMENTAL VARIABLES 6 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 - - - = 2 ? 2 Z Z Y Y Z Z b i i i ( 29 ( 29 ( 29 ( 29 - - - = 2 2 ? 2 Z Z X X Z Z b E i i i β ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - - = - - - - - - = - - - = X X Z Z Y Y Z Z Z Z Y Y Z Z X X Z Z Z Z b X X Z Z Z Z b i i i i i i i i i i i i i 2 2 ? 2 2 IV 2 However, we can neutralize the bias by multiplying b 2 ? By the reciprocal of the bias factor. We will call the new estimator b 2 IV , for reasons that will be explained later.
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INSTRUMENTAL VARIABLES 7 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i - + - = + + - + + = - 2 2 1 2 1 β β β β β ( 29 ( 29 ( 29 - - - = 2 ?
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