linear probability model 线性概率模型

LinearProbabilityModel(LPM)Example:WhatdetermineswhetheratownorcityhasastreetnamedafterMartinLutherKing?whereifzipcodehasastreetnamedafterMLKotherwiseHence,ResponseprobabilityLinearLinearProbabilityModelPrincipalAdvantageofLPM:easytointerpret,easytoexplain(.00063)[.00084]where(ordinarystandarderror)[robuststandarderror].A10percentagepointincreaseintheproportionofazipcode’sresidentswhoareAfrican-AmericanincreasestheprobabilitythatastreetwillbenamedafterMartinLutherKing,Jr.by4.7percentagepoints.Sinceonly8.5percentofzipcodesinFloridahaveastreetnamedafterhim,a4.7percentagepointincreaseissubstantial.DisadvantagesofLPM:1.Predictionsoutsideof0to1.2.Marginaleffectsareconstantunlessmodelisnon-linearinthevariables.3.Heteroskedasticerrors:StreetNamedAfterMLK,Jr.(1=yes)OrdinaryStandardErrorsHeteroskedasticRobustStandardErrors(1)(2)(3)(4)African-American(%)0.0047***0.0033***0.0047***0.0033***(9.13)(5.29)(5.84)(3.99)Urban0.0005**0.0005**(2.05)(2.24)Income(thou$)-0.0007-0.0007(0.85)(1.41)Poor(%)0.0038**0.0038**(2.45)(2.36)Constant0.0244**-0.01340.0244**-0.0134(2.21)(0.27)(2.53)(0.33)Observations910910910910R-squared0.0840.0990.0840.099Table1.ExplainingtheLikelihoodofNamingStreetsafterMartinLutherKing,Jr.Absolutevalueoft-statisticsinparentheses*significantat10%;**significantat5%;***significantat1%Non-LinearProbabilityModels(Probit&Logit)non-linearfunction^African-American(%)Non-LinearProbabilityModels(Probit&Logit)LatentVariable:ContinuousmeasureofthetendencytonameastreetafterMLK,Jr.Inotherwords,itisameasureofthelatentadmirationforMLK,Jr.TheoreticalStorymotivatingProbitandLogitConsiderseveralzipcodesinwhichalargeproportionoftheresidentsareAfrican-American.WhiletheyprobablyhaveahighlatentadmirationforMLK,Jr.,youcan’tbesurethatallofthemwillhavenamedastreetafterhim.000where0StreetNamedAfterMLK,Jr.(1=yes)LinearProbabilityModelProbitMeans(1)(2)(3)(4)MLKSt.(1=yes)0.085(0.278)African-American(%)12.780.0047***0.0033***0.0218***0.0137***(17.14)(5.84)(3.99)(7.56)(3.69)Urban(%)71.750.0005**0.0036*(38.05)(2.24)(1.93)Income(thousands)39.64-0.0007-0.0183*(13.89)(1.41)(1.87)Poor(%)13.040.0038**0.0109(8.74)(2.36)(1.00)Constant0.0244**-0.0134-1.7569***-1.3827***(2.53)(0.33)(20.65)(2.92)Observations910910910910910R-squared0.0840.099Absolutevalueoft-statisticsinparentheses*significantat10%;**significantat5%;***significantat1%African-American(%)12.780.01370.1751Urban(%)71.750.00360.2583Income(thousands)39.64-0.018-0.725Poor(%)13.040.01090.1421Constant1-1.383-1.383-1.5329𝑋𝛽𝛽∙𝑋𝛽∙𝑋=𝛽∙𝑋=−1.5329NORM.S.DIST(-1.5329,TRUE)=0.06265UsingEXCEL:African-American(%)13.780.01370.18879Urban(%)71.750.00360.2583Income(thousands)39.64-0.018-0.725Poor(%)13.040.01090.1421Constant1-1.383-1.383-1.5192𝑋𝛽𝛽∙𝑋𝛽∙𝑋=Add1%pointtoAfrican-American𝛽∙𝑋=−1.5192NORM.S.DIST(-1.5192,TRUE)=0.06436𝜕𝑝𝑟𝑜𝑏𝐾𝑖=1𝐴𝑖𝜕𝐴𝑖=0.06436−0.06265=0.0017^StreetNamedAfterMLK,Jr.(1=yes)LinearProbabilityModelProbitDerivativesMeans(1)(2)(3)(4)MLKSt.(1=yes)0.085(0.278)African-American(%)12.780.0047***0.0033***0.0029***0.0017***(17.14)(5.84)(3.99)(7.23)(3.54)Urban(%)71.750.0005**0.0004*(38.05)(2.24)(1.95)Income(thousands)39.64-0.0007-0.0023**(13.89)(1.41)(1.96)Poor(%)13.040.0038**0.0013(8.74)(2.36)(0.98)Constant0.0244**-0.0134(2.53)(0.33)Observations910910910910910R-squared0.0840.099Absolutevalueoft-statisticsinparentheses*significantat10%;**significantat5%;***significantat1%twoway(scatterMLKStreetAAm)(lineMLKStHatLPMAAm)(msplineMLKStHatLogitAAm)00.20.40.60.81020406080100𝐴At,theslopeoftheLPMisgreaterthantheLogit𝑨=𝟏𝟐.𝟖%“Whichcomesclosertoyourview?Gaycouplesshouldbeallowedtomarrylegally,ORgaycouplesshouldbeallowedtoformcivilunionsbutnotlegallymarry,ORthereshouldbenolegalrecognitionofagaycouple'srelationship?”InNovember,2009,theCBSNews/VanityFairMonthlyPollasked855adults:Forty-twopeopledidn’tanswerthequestionaboutgaymarriageandanother60weremissinginformationontheireducation,age,incomeorreligiousaffiliations,givingusafinalsampleof753people.Wejustlearnedabouttheprobitmodel,solet’sdefinethefollowingdichotomousvariable:whereifpersonirespondedthatgaysshouldbeallowedtolegallymarryotherwise𝐺𝑖=10GayMarriageDependentVariable:SupportLegalGayMarriage(1=yes)MeansLinearProbabilityModelProbitCoefficientsProbitDerivativesatMeans(1)(2)(3)SupportLegalGayMarriage(1=yes)0.368(0.483)College0.2140.0490.1570.0588(0.41)(1.23)(1.22)(1.2)GradDegree0.2030.151***0.474***0.1812***(0.403)(3.72)(3.64)(3.57)AGE54.788-0.002**-0.007**-0.0027**(17.079)(2.24)(2.23)(2.23)Female0.5980.057*0.179*0.0652*(0.491)(1.77)(1.70)(1.72)Athesist0.150.268***0.793***0.3062***(0.357)(5.95)(5.50)(5.6)Kids0.741-0.047-0.170-0.0634(0.438)(1.14)(1.28)(1.26)Ever_Married0.839-0.083-0.238-0.0901(0.367)(1.56)(1.42)(1.39)Republican0.316-0.129***-0.465***-0.1637***(0.465)(3.27)(3.43)(3.66)Democrat0.3520.245***0.696***0.2609***(0.478)(6.41)(5.81)(5.83)Constant0.436***-0.128(6.65)(0.61)Observations753753R-squared0.226PercentCorrectlyPredictedasaMeasureofGoodnessofFitDefinePredictMLKStreet=10𝑝𝑟𝑜𝑏𝐾𝑖=1𝑋𝑖0.5IfotherwiseDefinePredictSup