ln15 Tobits and Other Corner Solution Models.pdf

《ln15 Tobits and Other Corner Solution Models.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ApEc8212EconometricAnalysisII--Lecture#15TobitsandOtherCornerSolutionModelsReadings:Wooldridge,Chapter17(Sections1-8)I.IntroductionSometimesyouryvariablehitsanupperorlowerboundforalargenumberofobservations.Forexample,youmaybeinterestedinestimatingthedemandfortobaccoproducts,usingprices,income

2、andothervariablesasyourexplanatoryvariables.Yetmostpeopledon’tbuytobaccoproducts,soyourdependentvariable,expendituresontobaccoproducts,willbezeroformostofyourobservations.Wooldridgecallsmodelsforthesesituationscornersolutionmodels.Sometimestheyarecalled“censoredregressionmodels”,butinfactthed

3、ataarenotreallycensored.Wewilldiscuss“real”censoringmodelsinthenextlecture.Youmightask:What’swrongwithjustusinglinearregression(i.e.assumethatE[y

4、x]=x′β)whenfacedwithacornersolution?Theproblemsare:11.ItdoesnotmakesensethatE[y

5、x]islinearinx(partialeffectsareconstant)forvaluesofxthatleadmosty’s

6、tobeequaltozero.2.Forsomevaluesofx(andβ)youcanhaveE[y

7、x]<0,whichwouldnotmakesense.Oneapproachistotrytofixthisupwithcleverx′βfunctionalforms,e.g.E[y

8、x]=e.Thiswouldhavetobeestimatedusingnonlinearleastsquares(NLS).[Question:Whynotjusttakethelogofbothsides?]ButVar[y

9、x]islikelytobeheteroscedastic,

10、whichmeansNLSisnotefficient.Also,thisapproachdoesnotallowustoestimatesomerelationshipsofinterest,suchasP[y=0

11、x]andE[y

12、x,y>0].Toestimatesuchkindsofrelationshipsweneedtomakesomemoreassumptionsaboutthedistributionofyconditionalonx.ThestandardTypeITobitmodelhasthefollowingset-up:yi=max(0,xi′β+ui)

13、2ui

14、x~N(0,σ)2Notethatwiththenormalityassumptionwearespecifyingtheentiredistributionofyconditionalonx.Soanotherwaytoexpressthismodelis:2D(y

15、x)=Tobit(x′β,σ)Thisisjustnotation,whereD(y

16、x)denotesthedistributionofyconditionalonx.Sometimesitisusefultoexpressthisinlatentvariableform:2yi*=xi′β+ui,ui

17、

18、x~N(0,σ)yi(observedy=max(0,yi*)YetthiswayofwritingtheTobitmodelcouldbemisleadingifwearenotcareful,sincewereallyareinterestedinE[y

19、x],notE[y*

20、x]II.SomeVeryUsefulExpectedValueFormulasConsiderE[y

21、x]undertheassumptionsofaTypeITobitmodel.Assumetha