Variance with Heteroskedasticity For the general multiple regression model, a valid estimator of VarlB, with heterosked asticity is ∑ SsT2, Where r, is the i residual from regressing x, on all other independen t variable S, and SST, is the sum of squared residuals from this regression Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Variance with Heteroskedasticity ( ) ( ) is the sum of squared residuals from this regression regressing on all other independen t variable s, and , where ˆ is the residual from ˆ ˆ ˆ ˆ with heterosked asticity is ˆ estimator of For the general multiple regression model, a valid t h 2 2 j j i j j i j i j j SST x r i SST r u V ar Var b = b
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 o Typically call these robust standard errors Sometimes the estimated variance is corrected for degrees of freedom by multiplying by n/(n-k-1) ◆AsSn→∞ it's all the same, though Economics 20- Prof anderson 7
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 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) o 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 Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 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