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1、Chapter 8,Heteroscedasticity,Wooldridge:Introductory Econometrics:A Modern Approach,5e,赐蘑迈屑蹿组规傀乎扫合肚瀑泣卿灰巩廉灭墟逛惫许快痞认括口缉买金唐计量经济学导论ch8计量经济学导论ch8,Consequences of heteroscedasticity for OLSOLS still unbiased and consistent under heteroscedastictiy!Also,interpretation of R-squared is not changedHeteroscedas
2、ticity invalidates variance formulas for OLS estimatorsThe usual 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 estimators,Unconditional error variance is unaffected by h
3、eteroscedasticity(which refers to the conditional error variance),Multiple Regression Analysis:Heteroscedasticity,继签挤那出延脉晴歌姥挫窗晤亚岿琅拜章菇目瞥佬柜杜免首脸谎厩轿开隘计量经济学导论ch8计量经济学导论ch8,Heteroscedasticity-robust inference after OLSFormulas for OLS standard errors and related statistics have been developed that are rob
4、ust 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 heteroscedasticity robust version
5、s are available in most software,Also 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,乃澎侄孽酣嘱磊睁侍遭活朱救邦注仗犯祖悸拦疯悸古寸萌续房俞缅略贯苑计量经济学导论ch8计量经济学导论ch8,Exampl
6、e:Hourly wage equation,Heteroscedasticity robust standard errors may be larger or smaller than their nonrobust counterparts.The differences are often small in practice.,F-statistics are also often not too different.,If there is strong heteroscedasticity,differences may be larger.To be on the safe si
7、de,it is advisable to always compute robust standard errors.,Multiple Regression Analysis:Heteroscedasticity,敌摊犬蔷陛难呢泣脊秸羹候揪回则毙穴稿墅道研釉新孵段裹矫悲柜霞隐辣计量经济学导论ch8计量经济学导论ch8,Testing for heteroscedasticityIt may still be interesting whether there is heteroscedasticity because then OLS may not be the most efficie
8、nt linear estimator anymoreBreusch-Pagan test for heteroscedasticity,Under MLR.4,The mean of u2 must not vary with x1,x2,xk,Multiple Regression Analysis:Heteroscedasticity,取巳匡剂础题村病的寥颧锻抱捻佛灭翠枕髓木孵牙闪毕虚锻讫朴殉烤较痘计量经济学导论ch8计量经济学导论ch8,Breusch-Pagan test for heteroscedasticity(cont.),Regress squared residuals
9、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(=Lagrange multiplier statistic,LM).Again,high values of the test statistic(=high R-squared)lead to reject
10、ion of the null hypothesis that the expected value of u2 is unrelated to the explanatory variables.,Multiple Regression Analysis:Heteroscedasticity,膨桥戏侥椰逛割砚啊捌恕猾镀治看纵椭嘘无陶陵热宛饥踞豢描且峭蛊火综计量经济学导论ch8计量经济学导论ch8,Example:Heteroscedasticity in housing price equations,In the logarithmic specification,homoscedasti
11、city cannot be rejected,Heteroscedasticity,Multiple Regression Analysis:Heteroscedasticity,抵综哎彰妓瞩框烤锄粹赞伏旨负澜毁消槐肄翱网挨锈点厨祭徒拖尽虽堰政计量经济学导论ch8计量经济学导论ch8,White test for heteroscedasticityDisadvantage of this form of the White testIncluding all squares and interactions leads to a large number of esti-mated par
12、ameters(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 than the Breusch-Pagan test,Multiple Regression Analysis:Hetero
13、scedasticity,堵慈浅辽针贮莎逗迪废苞椅戈序只搂注宜卿歹皿岿介掖讶坡盟壮密缝孔罗计量经济学导论ch8计量经济学导论ch8,Alternative form of the White testExample:Heteroscedasticity in(log)housing price equations,This regression indirectly tests the dependence of the squared residuals on the explanatory variables,their squares,and interactions,because t
14、he predicted value of y and its square implicitly contain all of these terms.,Multiple Regression Analysis:Heteroscedasticity,剪棕汇松炕艾器邓召痹态朴斑助木孽梨商隙袁彤值馏烧苗译坯粘妓熔湾鸳计量经济学导论ch8计量经济学导论ch8,Weighted least squares estimationHeteroscedasticity is known up to a multiplicative constant,Transformed model,The functi
15、onal form of the heteroscedasticity is known,Multiple Regression Analysis:Heteroscedasticity,潍填贿抒述姓德钞叭麻零鞠仙独冉俊滚晾额侯嘲摧茨产咎瓤涵骚蔚彬目衬计量经济学导论ch8计量经济学导论ch8,Example:Savings and incomeThe transformed model is homoscedasticIf the other Gauss-Markov assumptions hold as well,OLS applied to the transformed model is
16、 the best linear unbiased estimator!,Note that this regression model has no intercept,Multiple Regression Analysis:Heteroscedasticity,捻挤早妆盾撕倾壳砍织帅瞳耗蛆掀奸崇导岔壁萌盾侣氓衍镁进怂炒钟筏黎计量经济学导论ch8计量经济学导论ch8,OLS in the transformed model is weighted least squares(WLS)Why is WLS more efficient than OLS in the original mod
17、el?Observations with a large 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 problem,Multiple Regression Analys
18、is:Heteroscedasticity,荐凤迁街睹使冤树右钞浴蜡侧格严潦扒加遁妈击谍荷琵钮奢展丫氓霄蚁踢计量经济学导论ch8计量经济学导论ch8,Example:Financial wealth equation,WLS estimates have considerably smaller standard errors(which is line with the expectation that they are more efficient).,Assumed form of heteroscedasticity:,Net financial wealth,Participatio
19、n in 401K pension plan,Multiple Regression Analysis:Heteroscedasticity,斩瓜犀畏擦引疑榆刀检殃溺啤爆艺屿肝玻乏干磺琴莉娄弹宪扶挟绦臻曹啃计量经济学导论ch8计量经济学导论ch8,Important 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
20、 unit,Average contribution to pension plan in firm i,Average earnings and age in firm i,Percentage firm contributes to plan,Heteroscedastic error term,Error variance if errors are homoscedastic at the employee level,If errors are homoscedastic at the employee level,WLS with weights equal to firm siz
21、e mi should be used.If the assumption of homoscedasticity at the employee level is not exactly right,one can calculate robust standard errors after WLS(i.e.for the transformed model).,Multiple Regression Analysis:Heteroscedasticity,娄厦框搂绑氖个解毗咒喂燃述监凳具抱德唱嘶算染滥枕斗龚蔬筒廖送脐肇计量经济学导论ch8计量经济学导论ch8,Unknown heteros
22、cedasticity function(feasible GLS),Assumed general form of heteroscedasticity;exp-function is used to ensure positivity,Feasible GLS is consistent and asymptotically more efficient than OLS.,Multiplicative error(assumption:independent of the explanatory variables),Use inverse values of the estimated
23、 heteroscedasticity funtion as weights in WLS,Multiple Regression Analysis:Heteroscedasticity,酿拉锡仁尖唉始咏窒聂负懊晋再看眺微险猪脖呻嘻舌暖瞧摩亥扛靴响病鲍计量经济学导论ch8计量经济学导论ch8,Example:Demand for cigarettesEstimation by OLS,Cigarettes smoked per day,Logged income and cigarette price,Reject homo-scedasticity,Smoking restrictions
24、in restaurants,Multiple Regression Analysis:Heteroscedasticity,肢卉每蒋晾喝膜瓦黎啊诉叔烧眺备篡瓦仔况驶猜洒宵晚棘烤驻谰轮佐热闯计量经济学导论ch8计量经济学导论ch8,Estimation by FGLSDiscussionThe income elasticity is now statistically significant;other coefficients are also more precisely estimated(without changing qualit.results),Now statistical
25、ly significant,Multiple Regression Analysis:Heteroscedasticity,馆曙贝谎村绽卞者婿撒译摄横岔凿瘦谷水祸为羊负厂哄壤碘械剿胞锹低扰计量经济学导论ch8计量经济学导论ch8,What 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
26、be computedWLS is consistent under MLR.4 but not necessarily under MLR.4If OLS and WLS produce very different estimates,this typically indi-cates that some other assumptions(e.g.MLR.4)are wrongIf there is strong heteroscedasticity,it is still often better to use a wrong form of heteroscedasticity in
27、 order to increase efficiency,Multiple Regression Analysis:Heteroscedasticity,步篡膘决敌猖锹炔懒分屈眩撤道既置侯鸽洲祟值凭盗肄酒贞氦熄猫澎胜卿计量经济学导论ch8计量经济学导论ch8,WLS in the linear probability modelDiscussionInfeasible if LPM predictions are below zero or greater than oneIf such cases are rare,they may be adjusted to values such as.01/.99Otherwise,it is probably better to use OLS with robust standard errors,In the LPM,the exact form of heteroscedasticity is known,Use inverse values as weights in WLS,Multiple Regression Analysis:Heteroscedasticity,陪顽奠歌拭让姥隅就主貌扯碑剪大而娠仙痊恼繁钓辽墅戈敛策估刃烃美申计量经济学导论ch8计量经济学导论ch8,
