Multiple regression analysis y-Bo+ Bx+B2x2+.. Bkrk+u ◆6. Heteroskedastici Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 6. Heteroskedasticity
What is Heteroskedasticity o Recall the assumption of homoskedasticity implied that conditional on the explanatory variables the variance of the unobserved error u was constant o 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
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
Example of Heteroskedasticity ECx)=Bo+ Bx X X X Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 . x1 x2 x f(y|x) Example of Heteroskedasticity x3 . . E(y|x) = b0 + b1x
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 o 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 4
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
Variance with Heteroskedasticity for the simple case, B,=B,+ ∑(x-x)x ∑( 、)2,SO 1n)=2(=)3 ssT2 L, where SST-2(x-x) x a valid estimator for this when o is 2(x-x)u? where i are are the ois residual SST Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Variance with Heteroskedasticity ( ) ( ) ( ) ( ) ( ) ( ) , where ˆ are are the OLS residuals ˆ A valid estimator for this when is , where ˆ ,so ˆ For the simple case, 2 2 2 2 2 i 2 2 2 2 1 1 1 2 i x i i x i x i i i i i u SST x x u SST x x SST x x Var x x x x u − = − − = − − = + b b b