# l41 - Specification Error Omitted and Extraneous Variables...

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Specification Error: Omitted and Extraneous Variables—Page 1 Specification Error: Omitted and Extraneous Variables Omitted variable bias. Suppose that the “correct” model is y X X 1 1 2 2 If we estimate y a b X b X e 1 1 2 2 we know that E(b 1 ) = 1 and E(b 2 ) = 2 i.e. the regression coefficients are unbiased estimators of the population parameters. Suppose, however, the researcher mistakenly believes y X * * * 1 1 and therefore estimates y a b X e * * * 1 1 i.e. X2 is mistakenly omitted from the model. How does b 1 (the regression estimate from the correctly specified model) compare to b 1 * (the regression estimate from the mis-specified model)? What is E(b 1 *)? Is it a biased or unbiased estimator of 1 ? If biased, how is it biased? Note that b 1 * ( , ) ( ) Cov X Y V X 1 1 Formula for bivariate regression coefficient ( , ) ( ) Cov X a b X b X e V X 1 1 1 2 2 1 Substitute the formula for Y from the correctly specified model ( , ) ( , ) ( , ) ( , ) ( ) Cov X a b Cov X X b Cov X X Cov X e V X 1 1 1 1 2 1 2 1 1 Expectations rules: Cov(a+b,c+d) = Cov(a,c) + Cov(a,d) + Cov(b,c) + Cov(b,d) 0 0 1 1 2 1 2 1 bV X b Cov X X V X ( ) ( , ) ( ) Recall that Cov(variable, constant) = 0. Also, X’s are uncorrelated with the residuals. b b Cov X X V X 1 2 1 2 1 ( , ) ( ) Simplify expression. Taking expectations, we get E b ( *) 1 1 2 12 1 2

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Specification Error: Omitted and Extraneous Variables—Page 2 Hence, b 1 * is a biased estimator of 1 . Further, this bias will not disappear as sample size gets larger, so the omission of a variable from a model also leads to an inconsistent estimator. Note that there are two conditions under which b
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## This note was uploaded on 02/29/2012 for the course SOC 63993 taught by Professor Richardwilliams during the Spring '11 term at Notre Dame.

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l41 - Specification Error Omitted and Extraneous Variables...

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