Implications for Standard Errors How then can we estimate the variance? The trick, a clever one, is to estimate with . That is, the squared residual for each observation “i” is used to estimate the population residual for observation “i”. • 13
Implications for Standard Errors The squared residual is not a very good estimator after all only one observation is used to estimate each observation’s variance, but it turns out to be good enough. The estimator is which provides a consistent estimate of in the presence of “arbitrary” heteroskedasticity. • 14
Implications for Standard Errors (MLR case) For comparison recall the earlier derivation leading to Thus This is the form we introduced earlier. • 15
Implications for Standard Errors (MLR case) Intuition is the same and the formula is residual for the i th observation from regressing x j on the other independent variables sum of squared residuals from the same regression • 16
Using Robust Standard Errors The robust standard error defined above have many names in the literature • White Std. Errrors • Eicker-White Std Errors • Huber SE • Robust SE • Heteroskedastic consistent standard errors, • etc. etc. etc. Use the one’s in bold – it is easier to call them what they are. 17
Using Robust Standard Errors Note: • Robust standard errors do not “correct” for heteroskedasticity, and • They are not adjusted versions of the original OLS standard errors. These are a different variance estimator that happens to be consistent in the presence of heteroskedasticity. 18
Using Robust Standard Errors Finally, Reporting robust standard errors does not affect coefficient estimates in any way we just trade an inconsistent estimator of standard errors for a consistent one. Nothing has been done to account for types of heteroskedasticity like known functions of X. 19
Using Robust Standard Errors So why not just report robust standard errors all the time? 1. In finite samples when the error term has a normal distribution, the usual OLS t-stats have an exact t-distribution. In contrast, the robust t-stats only have an asymptotic t-distribution thus if the data are really normally distributed the OLS standard errors will be more accurate and thus inference will be more accurate in finite samples. 20
Using Robust Standard Errors So why not just report robust standard errors all the time? 2. More generally, if the error terms is really homoscedastic there is an efficiency gain from imposing this assumption. The efficiency gain here comes from the variance of the estimates of the variance. • Remember that variance estimates themselves have a sampling distribution and we would like this sampling distribution to have a small variance, all else equal. 21
Using Robust Standard Errors, in practice Common to always report the robust standard error.
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- Fall '19
- Regression Analysis