C8D5 - INSTRUMENTAL VARIABLES 1 Suppose that you have a...

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Unformatted text preview: 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 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 ( 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 ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i- +- = + +- + + =- 2 2 1 2 1 ( 29 ( 29 ( 29 ( 29 --- = 2 2 ? 2 Z Z X X Z Z b E i i i INSTRUMENTAL VARIABLES 3 u X Y + + = 2 1 ( 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 ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i- +- = + +- + + =- 2 2 1 2 1 ( 29 ( 29 ( 29 ( 29 --- = 2 2 ? 2 Z Z X X Z Z b E i i i 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. INSTRUMENTAL VARIABLES 4 u X Y + + = 2 1 ( 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 ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i- +- = + +- + + =- 2 2 1 2 1 ( 29 ( 29 ( 29 ( 29 --- = 2 2 ? 2 Z Z X X Z Z b E i i i We decompose the expression. INSTRUMENTAL VARIABLES 5 u X Y + + = 2 1 ( 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 ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i- +- = + +- + + =- 2 2 1 2 1 ( 29 ( 29 ( 29 ( 29 --- = 2 2 ? 2 Z Z X X Z Z b E i i i 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 2 . INSTRUMENTAL VARIABLES 6 u X Y + + = 2 1 ( 29 ( 29 ( 29 u u X X u X u X Y Y i i i i i- +- = + +- + + =- 2 2 1 2 1 ( 29 ( 29 ( 29 --- = 2 ?...
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.

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C8D5 - INSTRUMENTAL VARIABLES 1 Suppose that you have a...

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