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Unformatted text preview: Contents 1 Introduction 1 1.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Sins of omission and inclusion 2 2.1 Omitted variable bias . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Irrelevance and inefficiency . . . . . . . . . . . . . . . . . . . . . 6 3 Other things 7 3.1 Proxy variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Testing linear restrictions . . . . . . . . . . . . . . . . . . . . . . 9 1 Introduction Contents 1.1 Model specification Contents • Basic question in econometrics: which regressors should be included, and how? • So far, have either assumed we know the ‘form’ of the true model ... – e.g. which X i appear as regressors • ...or skirted the issue altogether – e.g. we’ve seen various forms of ‘earnings regression’ in class – since there is only one ‘true’ model, at most one of these regressions was correctly specified What are the consequences of running a regression with the ‘wrong’ set of regressors? 1. If you omit a variable that should be there. . . • estimators of coefficients on included variables biased (in general) • standard errors for these coefficients incorrect (in general) 2. If you include a variable that shouldn’t be there. . . • estimators of coefficients on all variables unbiased but inefficient (in general) • standard errors on all coefficients correct (in general) • Will also look at role of proxy variables – variables that aren’t in the true model themselves, but are correlated with the true regressors – can be useful if we have no data on the true regressors • Will show how formally to test linear restrictions – e.g.any we impose to alleviate multicollinearity 2 Sins of omission and inclusion Contents 2.1 Omitted variable bias Contents Suppose the true model looks like this: Y = β 1 + β 2 X 2 + β 3 X 3 + u ∼ true model (6.1) But we don’t know X 3 is important and think it looks like this: Y = β 1 + β 2 X 2 + u ∼ perceived model (6.2) If we run OLS on (6.2) to get ˆ Y = b 1 + b 2 X 2 , problems... Y = β 1 + β 2 X 2 + β 3 X 3 + u ∼ true model (6.1) ˆ Y = b 1 + b 2 X 2 ∼ estimated model Problem #1: Bias The estimator b 2 is biased if X 2 and X 3 are correlated: E ( b 2 ) = β 2 + β 3 ∑ n i =1 ( X 2 i ¯ X 2 )( X 3 i ¯ X 3 ) ∑ n i =1 ( X 2 i ¯ X 2 ) 2 (6.5) = β 2 + β 3 r X 2 X 3 v u u t ∑ n i =1 ( X 3 i ¯ X 3 ) 2 ∑ n i =1 ( X 2 i ¯ X 2 ) 2 ) (proof p.202) 2 We can write the expression for E ( b 2 ) in terms of the sample correlation coefficient between X 2 and X 3 : E ( b 2 ) = β 2 + β 3 r X 2 X 3 v u u t ∑ n i =1 ( X 3 i ¯ X 3 ) 2 ∑ n i =1 ( X 2 i ¯ X 2 ) 2 • If β 3 and r X 2 X 3 have the same sign then b 2 is biased upwards • If β 3 and r X 2 X 3 have different signs then...
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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.
 Spring '08
 Witte
 Macroeconomics

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