chapter 6 [article]

chapter 6 [article] - Contents 1 Introduction 1 1.1 Model...

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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. weve 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 shouldnt 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 arent 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 = 1 + 2 X 2 + 3 X 3 + u true model (6.1) But we dont know X 3 is important and think it looks like this: Y = 1 + 2 X 2 + u perceived model (6.2) If we run OLS on (6.2) to get Y = b 1 + b 2 X 2 , problems... Y = 1 + 2 X 2 + 3 X 3 + u true model (6.1) 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 ) = 2 + 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) = 2 + 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 ) = 2 + 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 3 and r X 2 X 3 have the same sign then b 2 is biased upwards If 3 and r X 2 X 3 have different signs then...
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chapter 6 [article] - Contents 1 Introduction 1 1.1 Model...

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