计量经济学导论ch8课件

1、计量经济学导论ch8课件计量经济学导论ch8课件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 usual F-tests and t-tests are not valid under heterosced

2、asticity 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 variance) Multiple Regression Analysis: Heteroscedastici

3、tyConsequences of heteroscedastiHeteroscedasticity-robust inference after OLSFormulas for OLS standard errors and related statistics have been developed that are robust to heteroscedasticity of unknown formAll formulas are only valid in large samplesFormula for heteroscedasticity-robust OLS standard

4、 errorUsing these formulas, the usual t-test is valid asymptoticallyThe usual F-statistic does not work under heteroscedasticity, but heteroscedasticity robust versions are available in most softwareAlso called White/Eicker standard errors. They involve the squared residuals from the regression and

5、from a regression of xj on all other explanatory variables.Multiple Regression Analysis: HeteroscedasticityHeteroscedasticity-robust infeExample: Hourly wage equationHeteroscedasticity robust standard errors may be larger or smaller than their nonrobust counterparts. The differences are often small

6、in practice.F-statistics are 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: HeteroscedasticityExample: Hourly wage equationHTesting for heter

7、oscedasticityIt 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 must not vary with x1, x2, , xkMultiple Regression Analysis: HeteroscedasticityTestin

8、g for heteroscedasticityBreusch-Pagan test for heteroscedasticity (cont.)Regress squared residuals on all 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 (= La

9、grange multiplier statistic, LM). Again, high values of the test statistic (= high R-squared) lead to rejection of the null hypothesis that the expected value of u2 is unrelated to the explanatory variables.Multiple Regression Analysis: HeteroscedasticityBreusch-Pagan test for heterosExample: Hetero

10、scedasticity in housing price equationsIn the logarithmic specification, homoscedasticity cannot be rejectedHeteroscedasticityMultiple Regression Analysis: HeteroscedasticityExample: Heteroscedasticity inWhite test for heteroscedasticityDisadvantage of this form of the White testIncluding all square

11、s 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 in-teractions (here: example for k=3)The White test detects more general deviations from heteroscedasticity

12、 than the Breusch-Pagan testMultiple Regression Analysis: HeteroscedasticityWhite test for heteroscedasticAlternative form of the White testExample: Heteroscedasticity in (log) housing price equationsThis regression indirectly tests the dependence of the squared residuals on the explanatory variable

13、s, their squares, and interactions, because the predicted value of y and its square implicitly contain all of these terms. Multiple Regression Analysis: HeteroscedasticityAlternative form of the White Weighted least squares estimationHeteroscedasticity is known up to a multiplicative constantTransfo

14、rmed modelThe functional form of the heteroscedasticity is knownMultiple Regression Analysis: HeteroscedasticityWeighted least squares estimatExample: Savings and incomeThe transformed model is homoscedasticIf the other Gauss-Markov assumptions hold as well, OLS applied to the transformed model is t

15、he best linear unbiased estimator!Note that this regression model has no interceptMultiple Regression Analysis: HeteroscedasticityExample: Savings and incomeNotOLS in the transformed model is weighted least squares (WLS)Why is WLS more efficient than OLS in the original model?Observations with a lar

16、ge variance are less informative than observa-tions with small variance and therefore should get less weightWLS is a special case of generalized least squares (GLS)Observations with a large variance get a smaller weight in the optimization problemMultiple Regression Analysis: HeteroscedasticityOLS i

17、n the transformed model iExample: Financial wealth equationWLS estimates have considerably smaller standard errors (which is line with the expectation that they are more efficient).Assumed form of heteroscedasticity:Net financial wealthParticipation in 401K pension plan Multiple Regression Analysis:

18、 HeteroscedasticityExample: Financial wealth equaImportant special case of heteroscedasticityIf the observations are reported as averages at the city/county/state/-country/firm level, they should be weighted by the size of the unitAverage contribution to pension plan in firm iAverage earnings and ag

19、e in firm iPercentage firm contributes to planHeteroscedastic error termError variance if errors are homoscedastic at the employee levelIf errors are homoscedastic at the employee level, WLS with weights equal to firm size mi should be used. If the assumption of homoscedasticity at the employee leve

20、l is not exactly right, one can calculate robust standard errors after WLS (i.e. for the transformed model).Multiple Regression Analysis: HeteroscedasticityImportant special case of heteUnknown heteroscedasticity function (feasible GLS)Assumed general form of heteroscedasticity; exp-function is used

21、 to ensure positivityFeasible GLS is consistent and asymptotically more efficient than OLS.Multiplicative error (assumption: independent of the explanatory variables)Use inverse values of the estimated heteroscedasticity funtion as weights in WLSMultiple Regression Analysis: HeteroscedasticityUnknow

22、n heteroscedasticity funExample: Demand for cigarettesEstimation by OLSCigarettes smoked per dayLogged income and cigarette priceReject homo-scedasticitySmoking restrictions in restaurantsMultiple Regression Analysis: HeteroscedasticityExample: Demand for cigarettesEstimation by FGLSDiscussionThe in

23、come elasticity is now statistically significant; other coefficients are also more precisely estimated (without changing qualit. results)Now statistically significantMultiple Regression Analysis: HeteroscedasticityEstimation by FGLSNow statistiWhat if the assumed heteroscedasticity function is wrong?If the heteroscedasticity function is misspecified, WLS is still consistent under MLR.1 MLR.4, but robust standard errors should be computedWLS is consistent under MLR.4 but not neces