P kolm 38 variance of the ols estimators with

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Unformatted text preview: en by ˆˆ ru ˆ )= ∑ Var (β j 22 ij i 2 j SSR where ˆ • rij is the i th residual from regressing x j on all other independent variables • SSRj is the sum of squared residuals from this regression VER. 10/23/2012. © P. KOLM 38 Variance of the OLS Estimators with Heteroscedasticity (3/3) Main results: ˆˆ ∑r u ˆ • Var (β ) = j 22 ij i 2 j SSR ˆ is a consistent estimator of the variance of β j • The square root of this estimator can be used as the estimate for the standard error needed to perform inference • These standard errors are called robust or heteroscedasticity-robust standard errors, or White, Huber or Eicker standard errors • Most statistical packages can calculate these standard errors o Excel does not! VER. 10/23/2012. © P. KOLM 39 Remarks • A sensible question: If the heteroscedasticity-robust standard errors are valid more often than the usual OLS standard errors, why do we bother with the usual standard errors at all? o One reason the usual standard errors are still used in cross-sectional work is that, if the homoscedasticity assumption holds and the errors are normally distributed, then the usual t-statistics have exact tdistributions, regardless of the sample size o The robust standard errors and robust t-statistics are justified only as the sample size becomes large. With small sample sizes, the robust tstatistics can have distributions that are not very close to the tdistribution, and that could throw off our inference • In large sample sizes, we can make a case for always reporting only the heteroscedasticity-robust standard errors in cross-sectional applications. It is also common to report both standard errors so that a reader can determine whether any conclusions are sensitive to the standard error in use VER. 10/23/2012. © P. KOLM 40 • One can obtain F-statistics that are robust to heteroscedasticity of an unknown, arbitrary form. This version of the F-statistic is also called a heteroscedasticity-robust Wald statistic • A general treatment of the robust versions of the different statistics requires matrix algebra VER. 10/23/2012. © P. KOLM 41 Example (C8.2 in Wooldridge) (i) Use the data in HPRICE1.RAW to obtain the heteroscedasticity-robust standard errors for equation price = −21.77 + 0.002068lotsize + 0.1228sqrft + 13.85bdrms Discuss any important differences with the usual standard errors. Solution: The estimated equation with both sets of standard errors (heteroscedasticityrobust standard errors in brackets) is price = −21.77 + 0.002068lotsize + 0.1228sqrft + 13.85bdrms (29.48) (0.00064) (0.013) (9.01) [36.28] [0.00122] [0.017] [8.28] n = 88, R2 = 0.672. VER. 10/23/2012. © P. KOLM 42 Ordinary Least-squares Estimates Dependent Variable = price R-squared = 0.6724 Rbar-squared = 0.6607 sigma^2 = 3580.0453 Durbin-Watson = 2.1098 Nobs, Nvars = 88, 4 *************************************************************** Variable Coefficient t-statistic t-probability intercept -21.770309 -0.738601 0.462208...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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