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1、第9章 异方差：检验与修正Heteroskedasticity:test and correctionContents Whats heteroskedasticity?Why worry about heteroskedasticity?How to test the heteroskedasticity?Corrections for heteroskedasticity?Whats heteroskedasticity?Recall the assumption of homoskedasticity implied that conditional on the explanatory
2、 variables,the variance of the unobserved error,u,was constantvar(u|X)=s2(homoskedasticity)If this is not true,that is if the variance of u is different for different values of the Xs,then the errors are heteroskedasticvar(ui|Xi)=si2(heteroskedasticity).X1X2E(Y|X)=b0+b1XYf(Y|X)homoskedasticity.X3Exa
3、mple of Heteroskedasticity.X X1X2Yf(Y|X)X3.E(Y|X)=b0+b1XGenerally,cross-section data more easily induce heteroskedasticity because of different characteristics of different individuals.Consider a cross-section study of family income and expenditures.It seems plausible to expect that low income indiv
4、iduals would spend at a rather steady rate,while the spending patterns of high income families would be relatively volatile.If we examine sales of a cross section of firms in one industry,error terms associated with very large firms might have larger variances than those error terms associated with
5、smaller firms;sales of larger firms might be more volatile than sales of smaller firms.XYhomoskedasticity XYIncreasing with XXYComplicated heteroskedasticity YDecreasing with Xindindsalessalesrdexprdexpprofitprofitindindsalessalesrdexprdexpprofitprofitpackingpacking6375.36375.362.562.5185.1185.1nurs
6、enurse80552.880552.86620.16620.113869.913869.9nonbanknonbank11626.411626.492.992.91569.51569.5spacespace95294952943918.63918.64487.84487.8serviceservice14655.114655.1178.3178.3274.8274.8consumptionconsumption101314.1101314.11595.31595.310278.910278.9metalmetal21896.221896.2258.4258.42828.12828.1elec
7、tronicselectronics116141.3116141.36107.56107.58787.38787.3househouse26408.326408.3494.7494.7225.9225.9chemistrychemistry122315.7122315.74454.14454.116438.816438.8manufacturemanufacture32405.632405.6108310833751.93751.9polymerpolymer141649.9141649.93163.83163.89761.49761.4leisureleisure35107.735107.7
8、1620.61620.62884.12884.1computercomputer175025.8175025.813210.713210.719774.519774.5paperpaper40295.440295.4421.7421.74645.74645.7fuelfuel230614.5230614.51703.81703.822626.622626.6foodfood70761.670761.6509.2509.25036.45036.4autoauto2935432935439528.29528.218415.418415.4050001000015000R&D expenditure
9、(million dollars)0100000200000300000sales(million dollars)050001000015000R&D expenditure(million dollars)/Fitted values0100000200000300000sales(million dollars)Why Worry About Heteroskedasticity?The consequences of heteroskedasticity OLS estimates are still unbiased and consistent,even if we do not
10、assume homoskedasticity.take the simple regression as an example Y=b0+b1 X+uWe know the OLS estimator of b1 is 11221112iiiiiiiiiXX YXX uXXXXXX uEEXXbbbbb+The consequences of heteroskedasticity,cont.The R2 and adj-R2 are unaffected by heteroskedasticity.Because RSS and TSS are not affected by heteros
11、kedasticity,our R2 and adj-R2 are also not affected by heteroskedasticity.221111ESSRSSRTSSTSSRSSnkRTSSn The consequences of heteroskedasticity,cont.The standard errors of the estimates are biased if we have heteroskedasticity211112222222122var,varvarBecause of heteroskedasticity,then var,which are n
12、ot constant,therefore,var.However,OLS esiiiiiiiiiiiiiiXX uXX uXXuXXXXXXuXXXXbbbbssb+212timate of the variance of is.So,in this case,OLS estimates of the variances of the partial coefficients are biased.iXXsbThe consequences of heteroskedasticity,cont.The OLS estimates arent efficient,thats the varia
13、nces of the estimates are not the smallest variances.If the standard errors are biased,we can not use the usual t statistics or F statistics for drawing inferences.That is,the t test and F test and the confidence interval based on these test dont work.In a word,when there exists heteroskedasticity,w
14、e can not use t test and F test as usual.Or else,well get the misleading result.Summary of the consequences of heteroskedasticity OLS estimates are still unbiased and consistent The R2 and adj-R2 are unaffected by heteroskedasticity The standard errors of the estimates are biased.The OLS estimates a
15、rent efficient.Then,the t test and F test and the confidence interval dont work.How to test the heteroskedasticity?Residual plot w In the OLS estimation,we often use the residual ei to estimate the random error term ui,therefore,we can test whether there is heteroskedasticity of ui by examine ei.We
16、plot the scatter graph between ei2 and X.Residual plot,cont.Xe2a)homoskedasticity Xe2b)Xe2c)Xe2d)Xe2e)Residual plot,cont.w If there are more than one independent variables,we should plot the residual squared with all the independent variables,separately.w There is a shortcut to do the residual plot test when there are more than 1 independent variables.That is,we plot the residual with the fitted value,because is just the linear combination of all Xs.Residual plot:example 9.2-50000500010000Residu
