# ch08 - Economics 20 Prof Anderson 1 Multiple Regression...

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Unformatted text preview: Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = β + β 1 x 1 + β 2 x 2 + . . . β k x k + u 6. Heteroskedasticity Economics 20 - Prof. Anderson 2 What is Heteroskedasticity Recall the assumption of homoskedasticity implied that conditional on the explanatory variables, the variance of the unobserved error, u , was constant If this is not true, that is if the variance of u is different for different values of the x ’s, then the errors are heteroskedastic Example: estimating returns to education and ability is unobservable, and think the variance in ability differs by educational attainment Economics 20 - Prof. Anderson 3 . x x 1 x 2 y f( y|x ) Example of Heteroskedasticity x 3 . . E( y | x ) = β + β 1 x Economics 20 - Prof. Anderson 4 Why Worry About Heteroskedasticity? OLS is still unbiased and consistent, even if we do not assume homoskedasticity The standard errors of the estimates are biased if we have heteroskedasticity If the standard errors are biased, we can not use the usual t statistics or F statistics or LM statistics for drawing inferences Economics 20 - Prof. Anderson 5 Variance with Heteroskedasticity ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 residuals OLS the are are ˆ where , ˆ is when for this estimator A valid where , ˆ so , ˆ case, simple For the 2 2 2 2 2 i 2 2 2 2 1 2 1 1 i x i i i x x i i i i i u SST u x x x x SST SST x x Var x x u x x ∑ ∑ ∑ ∑ ∑- ≠- =- =-- + = σ σ σ β β β Economics 20 - Prof. Anderson 6 Variance with Heteroskedasticity ( 29 ( 29 regression this from residuals squared of sum the is and s, t variable independen other all on regressing from residual the is ˆ where , ˆ ˆ ˆ ˆ is asticity heterosked with ˆ of estimator valid a model, regression multiple general For the th 2 2 j j ij j i ij j j SST x i r SST u r r Va Var ∑ = β β Economics 20 - Prof. Anderson 7 Robust Standard Errors Now that we have a consistent estimate of the variance, the square root can be used as...
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ch08 - Economics 20 Prof Anderson 1 Multiple Regression...

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