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1、 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Chapter 8 Heteroscedasticity Wooldridge:Introductory Econometrics:A Modern Approach,5e 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied
2、 or duplicated,or posted to a publicly accessible website,in whole or in part.Consequences of heteroscedasticity for OLSOLS still unbiased and consistent under heteroscedastictiy!Also,interpretation of R-squared is not changedHeteroscedasticity invalidates variance formulas for OLS estimatorsThe usu
3、al F-tests and t-tests are not valid under heteroscedasticity Under heteroscedasticity,OLS is no longer the best linear unbiased estimator(BLUE);there may be more efficient linear estimatorsUnconditional error variance is unaffected by heteroscedasticity(which refers to the conditional error varianc
4、e)Multiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Heteroscedasticity-robust inference after OLSFormulas for OLS standard errors and related statistics have
5、 been developed that are robust to heteroscedasticity of unknown formAll formulas are only valid in large samplesFormula for heteroscedasticity-robust OLS standard errorUsing these formulas,the usual t-test is valid asymptoticallyThe usual F-statistic does not work under heteroscedasticity,but heter
6、oscedasticity robust versions are available in most softwareAlso called White/Eicker standard errors.They involve the squared residuals from the regression and from a regression of xj on all other explanatory variables.Multiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights
7、Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Hourly wage equationHeteroscedasticity robust standard errors may be larger or smaller than their nonrobust counterparts.The differences are often small in practice.F-statistics ar
8、e also often not too different.If there is strong heteroscedasticity,differences may be larger.To be on the safe side,it is advisable to always compute robust standard errors.Multiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplic
9、ated,or posted to a publicly accessible website,in whole or in part.Testing for heteroscedasticityIt may still be interesting whether there is heteroscedasticity because then OLS may not be the most efficient linear estimator anymoreBreusch-Pagan test for heteroscedasticityUnder MLR.4The mean of u2
10、must not vary with x1,x2,xkMultiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Breusch-Pagan test for heteroscedasticity(cont.)Regress squared residuals on all
11、 expla-natory variables and test whether this regression has explanatory power.A large test statistic(=a high R-squared)is evidence against the null hypothesis.Alternative test statistic(=Lagrange multiplier statistic,LM).Again,high values of the test statistic(=high R-squared)lead to rejection of t
12、he null hypothesis that the expected value of u2 is unrelated to the explanatory variables.Multiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Heterosc
13、edasticity in housing price equationsIn the logarithmic specification,homoscedasticity cannot be rejectedHeteroscedasticityMultiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in w
14、hole or in part.White test for heteroscedasticityDisadvantage of this form of the White testIncluding all squares and interactions leads to a large number of esti-mated parameters(e.g.k=6 leads to 27 parameters to be estimated)Regress squared residuals on all expla-natory variables,their squares,and
15、 in-teractions(here:example for k=3)The White test detects more general deviations from heteroscedasticity than the Breusch-Pagan testMultiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible w
16、ebsite,in whole or in part.Alternative form of the White testExample:Heteroscedasticity in(log)housing price equationsThis regression indirectly tests the dependence of the squared residuals on the explanatory variables,their squares,and interactions,because the predicted value of y and its square implicitly contain all of these terms.Multiple Regression Analysis:Heteroscedasticity 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessibl
