计量经济学课件ch08.ppt

Economics 20 - Prof. Anderson 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 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 Variance with Heteroskedasticity Robust Standard Errors Now that we have a consistent estimate of the variance, the square root can be used as a standard error for inference Typically call these robust standard errors Sometimes the estimated variance is corrected for degrees of freedom by multiplying by n/(n – k – 1) As n → ∞ it’s all the same, though Robust Standard Errors (cont) Important to remember that these robust standard errors only have asymptotic justification – with small sample sizes t statistics formed with robust standard errors will not have a distribution close to the t, and inferences will not be correct In Stata, robust standard errors are easily obtained using the robust option of reg A Robust LM Statistic Run OLS on the restricted model and save the residuals ? Regress each of the excluded variables on all of the included variables (q different regressions) and save each set of residuals ?1, ?2, …, ?q Regress a variable defined to be = 1 on ?1 ?, ?2 ?, …, ?q ?, with no intercept The LM statistic is n – SSR1, where SSR1 is the sum of squared residuals from this final regression Testing for Heteroskedasticity Es