十讲 虚拟变量DUMMY VARIALBE

第十讲虚拟变量DUMMYVARIALBEDUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESDUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESTWOSETSOFDUMMYVARIABLESSLOPEDUMMYVARIABLESSIB-BFSU,ECONOMETRICS1LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThissequenceexplainshowyoucanincludequalitativeexplanatoryvariablesinyourregressionmodel.SIB--BFSU,ECONOMETRICS2LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESSupposethatyouhavedataontheannualrecurrentexpenditure,COST,andthenumberofstudentsenrolled,N,forasampleofsecondaryschools,ofwhichtherearetwotypes:regularandoccupational.SIB--BFSU,ECONOMETRICS3LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESTheoccupationalschoolsaimtoprovideskillsforspecificoccupationsandtheytendtoberelativelyexpensivetorunbecausetheyneedtomaintainspecializedworkshops..SIB--BFSU,ECONOMETRICS4LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESOnewayofdealingwiththedifferenceinthecostswouldbetorunseparateregressionsforthetwotypesofschool.SIB--BFSU,ECONOMETRICS5LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESHoweverthiswouldhavethedrawbackthatyouwouldberunningregressionswithtwosmallsamplesinsteadofonelargeone,withanadverseeffectontheprecisionoftheestimatesofthecoefficients.SIB--BFSU,ECONOMETRICS6LECTURE10OCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1'+b2N+uDUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESAnotherwayofhandlingthedifferencewouldbetohypothesizethatthecostfunctionforoccupationalschoolshasaninterceptb1'thatisgreaterthanthatforregularschools.b1b1'SIB--BFSU,ECONOMETRICS7LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESEffectively,wearehypothesizingthattheannualoverheadcostisdifferentforthetwotypesofschool,butthemarginalcostisthesame.Themarginalcostassumptionisnotveryplausibleandwewillrelaxitinduecourse.OCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1'+b2N+ub1b1'SIB--BFSU,ECONOMETRICS8LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESdLetusdefinedtobethedifferenceintheintercepts:d=b1'-b1.OCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1'+b2N+ub1b1'SIB--BFSU,ECONOMETRICS9LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThenb1'=b1+dandwecanrewritethecostfunctionforoccupationalschoolsasshown.b1+ddOCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1+d+b2N+ub1SIB--BFSU,ECONOMETRICS10LECTURE10CombinedequationCOST=b1+dOCC+b2N+uOCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1+d+b2N+uDUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWecannowcombinethetwocostfunctionsbydefiningadummyvariableOCCthathasvalue0forregularschoolsand1foroccupationalschools.db1b1+dSIB--BFSU,ECONOMETRICS11LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESDummyvariablesalwayshavetwovalues,0or1.IfOCCisequalto0,thecostfunctionbecomesthatforregularschools.IfOCCisequalto1,thecostfunctionbecomesthatforoccupationalschools.db1b1+dCombinedequationCOST=b1+dOCC+b2N+uOCC=0RegularschoolCOST=b1+b2N+uOCC=1OccupationalschoolCOST=b1+d+b2N+uSIB--BFSU,ECONOMETRICS12LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWewillnowfitafunctionofthistypeusingactualdataforasampleof74secondaryschoolsinShanghai.SIB--BFSU,ECONOMETRICS13LECTURE10SchoolTypeCOSTNOCC1 Occupational345,,000 62312 Occupational537,00065313 Regular170,000400 04 Occupational526.00066315 Regular100,00056306 Regular28,00023607 Regular160,000307 08 Occupational45,,000 17319 Occupational120,000146110Occupational61,000991DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThetableshowsthedataforthefirst10schoolsinthesample..Theannualcostismeasuredinyuan,oneyuanbeingworthabout20centsU.S.atthetime.Nisthenumberofstudentsintheschool..SIB--BFSU,ECONOMETRICS14LECTURE10SchoolTypeCOSTNOCC1 Occupational345,,000 62312 Occupational537,00065313 Regular170,000400 04 Occupational526.00066315 Regular100,00056306 Regular28,00023607 Regular160,000307 08 Occupational45,,000 17319 Occupational120,000146110Occupational61,000991DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESOCCisthedummyvariableforthetypeofschool.SIB--BFSU,ECONOMETRICS15LECTURE10.DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::19:22Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C -33612.5523573.47--1.4258640.1583N 331.4493 39.75844 8..336578 0..0000OCC133259.120827..596.3982010.0000R-squared0.615637Meandependentvar187418.0AdjustedR-squared0.604810S..D.dependentvar141969.9S.E..ofregression89248..09Akaikeinfocriterion25..67592Sumsquaredresid5.66E+11Schwarzcriterion25.76933Loglikelihood--947.0092F--statistic56.86072Durbin--Watsonstat2.422989Prob((F-statistic))0.000000DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWenowruntheregressionofCOSTonNandOCC,treatingOCCjustlikeanyotherexplanatoryvariable,despiteitsartificialnature.SIB--BFSU,ECONOMETRICS16LECTURE10

DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESCOST=-34,000++133,,000OCC+331N^Theregressionresultshavebeenrewritteninequationform.FromitwecanderivecostfunctionsforthetwotypesofschoolbysettingOCCequalto0or1..SIB--BFSU,ECONOMETRICS17LECTURE10

RegularSchool(OCC=0))DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESCOST=-34,000++133,,000OCC+331NCOST=-34,000++331N^^IfOCCisequalto0,wegettheequationforregularschools,,asshown.Itimpliesthatthemarginalcostperstudentperyearis331yuanandthattheannualoverheadcostis--34,,000yuan..SIB--BFSU,ECONOMETRICS18LECTURE10

RegularSchool(OCC=0))DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESCOST=-34,000++133,,000OCC+331NCOST=-34,000++331N^^Obviouslyhavinganegativeinterceptdoesnotmakeanysenseatallanditsuggeststhatthemodelismisspecifiedinsomeway.Wewillcomebacktothislater.SIB--BFSU,ECONOMETRICS19LECTURE10

RegularSchool(OCC=0))OccupationalSchool(OCC=1))DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESPuttingOCCequalto1,weestimatetheannualoverheadcostofanoccupationalschooltobe99,000yuan.Themarginalcostisthesameasforregularschools..Itmustbe,giventhemodelspecification.COST=-34,000++133,,000OCC+331NCOST=-34,000++331NCOST=-34,000++133,,000+331N=99,000++331N^^^SIB--BFSU,ECONOMETRICS20LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThescatterdiagramshowsthedataandthetwocostfunctionsderivedfromtheregressionresults.SIB--BFSU,ECONOMETRICS21LECTURE10.DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::19:22Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C -33612.5523573.47--1.4258640.1583N 331.4493 39.75844 8..336578 0..0000OCC133259.120827..596.3982010.0000R-squared0.615637Meandependentvar187418.0AdjustedR-squared0.604810S..D.dependentvar141969.9S.E..ofregression89248..09Akaikeinfocriterion25..67592Sumsquaredresid5.66E+11Schwarzcriterion25.76933Loglikelihood--947.0092F--statistic56.86072Durbin--Watsonstat2.422989Prob((F-statistic))0.000000DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWewillperformattestonthecoefficientofthedummyvariable..ournullhypothesisisthatthereisnodifferenceintheoverheadcostsofthetwotypesofschool.Thetstatisticis6.40,soitisrejectedatthe0..1%significancelevel.SIB--BFSU,ECONOMETRICS22LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThissequenceexplainshowtoextendthedummyvariabletechniquetohandleaqualitativeexplanatoryvariablewhichhasmorethantwocategories.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS23LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIntheprevioussequenceweusedadummyvariabletodifferentiatebetweenregularandoccupationalschoolswhenfittingacostfunction.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS24LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESInactualfacttherearetwotypesofregularsecondaryschoolinShanghai.Therearegeneralschools,,whichprovidetheusualacademiceducation,andvocationalschools.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS25LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESAstheirnameimplies,,thevocationalschoolsaremeanttoimpartoccupationalskillsaswellasgiveanacademiceducation..COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS26LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESHoweverthevocationalcomponentofthecurriculumistypicallyquitesmallandtheschoolsaresimilartothegeneralschools..Oftentheyarejustgeneralschoolswithacoupleofworkshopsadded.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS27LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESLikewisetherearetwotypesofoccupationalschool..Therearetechnicalschoolstrainingtechniciansandskilledworkers’schoolstrainingcraftsmen..COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS28LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESSonowthequalitativevariablehasfourcategories..Thestandardprocedureistochooseonecategoryasthereferencecategoryandtodefinedummyvariablesforeachoftheothers.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS29LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIngeneralitisgoodpracticetoselectthemostnormalorbasiccategoryasthereferencecategory,ifonecategoryisinsomesensemorenormalorbasicthantheothers..COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS30LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIntheShanghaisampleitissensibletochoosethegeneralschoolsasthereferencecategory..Theyarethemostnumerousandtheotherschoolsarevariationsofthem.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS31LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESAccordinglywewilldefinedummyvariablesfortheotherthreetypes.TECHwillbethedummyforthetechnicalschools:TECHisequalto1iftheobservationrelatestoatechnicalschool,0otherwise..COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS32LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESEachofthedummyvariableswillhaveacoefficientwhichrepresentstheextraoverheadcostsoftheschools,,relativetothereferencecategory..COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS33LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESNotethatyoudonotincludeadummyvariableforthereferencecategory,,andthatisthereasonthatthereferencecategoryisusuallydescribedastheomittedcategory.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uSIB--BFSU,ECONOMETRICS34LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIfanobservationrelatestoageneralschool,,thedummyvariablesareall0andtheregressionmodelisreducedtoitsbasiccomponents.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uGeneralSchoolCOST=b1+b2N+u(TECH=WORKER=VOC=0))SIB--BFSU,ECONOMETRICS35LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIfanobservationrelatestoatechnicalschool,TECHwillbeequalto1andtheotherdummyvariableswillbe0.Theregressionmodelsimplifiesasshown.COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uGeneralSchoolCOST=b1+b2N+u(TECH=WORKER=VOC=0))TechnicalSchoolCOST=((b1+dT)+b2N+u(TECH=1;;WORKER=VOC=0))SIB--BFSU,ECONOMETRICS36LECTURE10COST=b1+dTTECH+dWWORKER+dVVOC+b2N+uGeneralSchoolCOST=b1+b2N+u(TECH=WORKER=VOC=0))TechnicalSchoolCOST=((b1+dT)+b2N+u(TECH=1;;WORKER=VOC=0))SkilledWorkers’’SchoolCOST=((b1+dW)+b2N+u(WORKER=1;;TECH=VOC=0))VocationalSchoolCOST=((b1+dV)+b2N+u(VOC=1;;TECH=WORKER=0))DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESTheregressionmodelsimplifiesinasimilarmannerinthecaseofobservationsrelatingtoskilledworkers’schoolsandvocationalschools.SIB--BFSU,ECONOMETRICS37LECTURE10COSTNb1+dTb1+dWb1+dVb1Workers’VocationaldWdVdTDUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThediagramillustratesthemodelgraphically..Thedcoefficientsaretheextraoverheadcostsofrunningtechnical,skilledworkers’,andvocationalschools,relativetotheoverheadcostofgeneralschools.TechnicalGeneralSIB--BFSU,ECONOMETRICS38LECTURE10COSTNdWdVdTNotethatwedonotmakeanypriorassumptionaboutthesize,,oreventhesign,,ofthedcoefficients..Theywillbeestimatedfromthesampledata.DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESWorkers’VocationalTechnicalGeneralb1+dTb1+dWb1+dVb1SIB--BFSU,ECONOMETRICS39LECTURE10SchoolTypeCOSTNTECHWORKER VOC1 Technical345,,000 6231 002 Technical537,00065310 03 General170,000400 0004 Workers’’526..000 6630 105 General100,000563 0006 Vocational28,00023600 17 Vocational160,000307 0018 Technical45,,000 1731 009 Technical120,00014610 010Workers’’61,0009901 0Herearethedataforthefirst10ofthe74schools.NotehowthevaluesofthedummyvariablesTECH,WORKER,andVOCaredeterminedbythetypeofschoolineachobservation.DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESSIB--BFSU,ECONOMETRICS40LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThescatterdiagramshowsthedatafortheentiresample,differentiatingbytypeofschool.SIB--BFSU,ECONOMETRICS41LECTURE10DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::20:32Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C -54893.0926673.08--2.0579960.0434N 342.6335 40.21950 8..519090 0..0000TECH 154110.9 26760.41 5..758915 0..0000WORKER143362.427852..805.1471440.0000VOC53228..6431061..651.7136450.0911R-squared0.632050Meandependentvar187418.0AdjustedR-squared0.610719S..D.dependentvar141969.9S.E..ofregression88578..37Akaikeinfocriterion25..68634Sumsquaredresid5.41E+11Schwarzcriterion25.84202Loglikelihood--945.3946F--statistic29.63132Durbin--Watsonstat2.503728Prob((F-statistic))0.000000DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThecoefficientofNindicatesthatthemarginalcostperstudentperyearis343yuan.SIB--BFSU,ECONOMETRICS42LECTURE10DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::20:32Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C -54893.0926673.08--2.0579960.0434N 342.6335 40.21950 8..519090 0..0000TECH 154110.926760.415.7589150.0000WORKER143362.427852.805.1471440.0000VOC53228..6431061.651.7136450.0911R-squared0.632050Meandependentvar187418.0AdjustedR-squared0.610719S..D.dependentvar141969.9S.E..ofregression88578..37Akaikeinfocriterion25..68634Sumsquaredresid5.41E+11Schwarzcriterion25.84202Loglikelihood--945.3946F--statistic29.63132Durbin--Watsonstat2.503728Prob((F-statistic))0.000000DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThecoefficientsofTECH,WORKER,,andVOCare154,000,143,000,and53,000,respectively,andshouldbeinterpretedastheadditionalannualoverheadcosts,relativetothoseofgeneralschools.SIB--BFSU,ECONOMETRICS43LECTURE10COST=-55,,000+154,000TECH+143,000WORKER+53,000VOC+343NGeneralSchoolCOST=--55,,000+343N(TECH=WORKER=VOC=0))TechnicalSchoolCOST=-55,000++154,,000+343N(TECH=1;;WORKER=VOC=0))=99,000++343NSkilledWorkers’’SchoolCOST=-55,000++143,,000+343N(WORKER=1;;TECH=VOC=0))=88,000++343NVocationalSchoolCOST=-55,000++53,000++343N(VOC=1;;TECH=WORKER=0))=--2,000++343NDUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESNotethatineachcasetheannualmarginalcostperstudentisestimatedat343yuan..Themodelspecificationassumesthatthisfiguredoesnotdifferaccordingtotypeofschool.^^^^^SIB--BFSU,ECONOMETRICS44LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThefourcostfunctionsareillustratedgraphically..SIB--BFSU,ECONOMETRICS45LECTURE10TWOSETSOFDUMMYVARIABLESTheexplanatoryvariablesinaregressionmodelmayincludemultiplesetsofdummyvariables.Thissequenceprovidesanexampleofamodelwithtwotypes.COST=b1+dOCC+eRES+b2N+uSIB--BFSU,ECONOMETRICS46LECTURE10TWOSETSOFDUMMYVARIABLESWewillcontinuewiththeschoolcostfunctionmodelandextendittotakeaccountofthefactthatsomeoftheschoolsareresidential..COST=b1+dOCC+eRES+b2N+uSIB--BFSU,ECONOMETRICS47LECTURE10TWOSETSOFDUMMYVARIABLESTomodelthehigheroverheadcostsofresidentialschools,,weintroduceadummyvariableRESwhichisequalto1forthemand0fornon--residentialschools.eistheextraannualoverheadcostofaresidentialschool,,relativetothatofanon-residentialone.COST=b1+dOCC+eRES+b2N+uSIB--BFSU,ECONOMETRICS48LECTURE10TWOSETSOFDUMMYVARIABLESWewillalsomakeadistinctionbetweenoccupationalandregularschools,,usingthedummyvariableOCCdefinedinthefirstsequence.COST=b1+dOCC+eRES+b2N+uSIB--BFSU,ECONOMETRICS49LECTURE10TWOSETSOFDUMMYVARIABLESInthecaseofanon-residentialoccupationalschool,RESis0andOCCis1,sotheoverheadcostincreasesbyd.Iftheschoolisbothoccupationalandresidential,itincreasesby(d+e).COST=b1+dOCC+eRES+b2N+uRegular,non--residentialCOST=b1+b2N+u(OCC=RES=0))Regular,residentialCOST=((b1+e)+b2N+u(OCC=0;;RES=1))Occupational,,non-residentialCOST=((b1+d)+b2N+u(OCC=1;;RES=0))Occupational,,residentialCOST=((b1+d+e)+b2N+u(OCC=RES=1))SIB--BFSU,ECONOMETRICS50LECTURE10COSTNb1+d+eb1+db1+eb1Occupational,,residentialRegular,non--residentialded+eeOccupational,,non--residentialRegular,residentialTWOSETSOFDUMMYVARIABLESThediagramillustratesthemodelgraphically..Notethattheeffectsofthedifferentcomponentsofthemodelareassumedtobeseparateandadditiveinthisspecification.SIB--BFSU,ECONOMETRICS51LECTURE10TWOSETSOFDUMMYVARIABLESHerearethedataforthefirst10schools..Notehowthevaluesofthedummyvariablesvaryaccordingtothecharacteristicsoftheschool.SchoolTypeResidential?COSTNOCCRES1 OccupationalNo345,,000 6231 02 OccupationalYes 537,000653 113 RegularNo 170,000400 004 OccupationalYes 526.000663 115 RegularNo100,000563006 RegularNo 28,000236007 RegularYes160,,000 3070 18 OccupationalNo45,,000 1731 09 OccupationalNo120,0001461010OccupationalNo61,0009910SIB--BFSU,ECONOMETRICS52LECTURE10DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::21:06Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C -29045.2723291.54--1.2470310.2165N 321.8330 39.40225 8..167884 0..0000OCC109564.624039..584.5576740.0000RES57909..0130821..311.8788630.0644R-squared0.634090Meandependentvar187418.0AdjustedR-squared0.618409S..D.dependentvar141969.9S.E..ofregression87699..16Akaikeinfocriterion25..65375Sumsquaredresid5.38E+11Schwarzcriterion25.77829Loglikelihood--945.1888F--statistic40.43470Durbin--Watsonstat2.443610Prob((F-statistic))0.000000TWOSETSOFDUMMYVARIABLESWewillstartbyinterpretingtheregressioncoefficients..ThecoefficientofNindicatesthatthemarginalcostperstudentis322yuanperyear.SIB--BFSU,ECONOMETRICS53LECTURE10TWOSETSOFDUMMYVARIABLESCOST=-29,,000+110,000OCC+58,000RES+322NRegular,non--residentialCOST=--29,,000+322N(OCC=RES=0))Regular,residentialCOST=--29,,000+58,,000+322N(OCC=0;;RES=1))=29,000++322NOccupational,,non-residentialCOST=--29,,000+110,000++322N(OCC=1;;RES=0))=81,000++322NOccupational,,residentialCOST=--29,,000+110,000++58,000++322N(OCC=RES=1))=139,,000+322N^^^^^SIB--BFSU,ECONOMETRICS54LECTURE10SLOPEDUMMYVARIABLESThescatterdiagramshowsthedataforthe74schoolsinShanghaiandthecostfunctionsderivedfromaregressionofCOSTonNandadummyvariableforthetypeofcurriculum((occupational/regular)..SIB--BFSU,ECONOMETRICS55LECTURE10SLOPEDUMMYVARIABLESThespecificationofthemodelincorporatestheassumptionthatthemarginalcostperstudentisthesameforoccupationalandregularschools.Hencethecostfunctionsareparallel.SIB--BFSU,ECONOMETRICS56LECTURE10SLOPEDUMMYVARIABLESHowever,thisisnotarealisticassumption.Occupationalschoolsincurexpenditureontrainingmaterialsthatisrelatedtothenumberofstudents.SIB--BFSU,ECONOMETRICS57LECTURE10SLOPEDUMMYVARIABLESAlso,thestaff--studentratiohastobehigherinoccupationalschoolsbecauseworkshopgroupscannotbe,oratleastshouldnotbe,aslargeasacademicclasses.SIB--BFSU,ECONOMETRICS58LECTURE10SLOPEDUMMYVARIABLESLookingatthescatterdiagram,,youcanseethatthecostfunctionfortheoccupationalschoolsshouldbesteeper,andthatfortheregularschoolsshouldbeflatter..SIB--BFSU,ECONOMETRICS59LECTURE10SLOPEDUMMYVARIABLESWewillrelaxtheassumptionofthesamemarginalcostbyintroducingwhatisknownasaslopedummyvariable.ThisisNOCC,definedastheproductofNandOCC.COST=b1+dOCC+b2N+lNOCC+u

SIB--BFSU,ECONOMETRICS60LECTURE10SLOPEDUMMYVARIABLESInthecaseofaregularschool,OCCis0andhencesoalsoisNOCC.Themodelreducestoitsbasiccomponents.COST=b1+dOCC+b2N+lNOCC+uRegularschoolCOST=b1+b2N+u(OCC=NOCC=0))

SIB--BFSU,ECONOMETRICS61LECTURE10SLOPEDUMMYVARIABLESInthecaseofanoccupationalschool,OCCisequalto1andNOCCisequaltoN.Theequationsimplifiesasshown.COST=b1+dOCC+b2N+lNOCC+uRegularschoolCOST=b1+b2N+u(OCC=NOCC=0))OccupationalschoolCOST=((b1+d)+((b2+l)N+u(OCC=1;;NOCC=N)SIB--BFSU,ECONOMETRICS62LECTURE10SLOPEDUMMYVARIABLESThemodelnowallowsthemarginalcostperstudenttobeanamountlgreaterthanthatinregularschools,,aswellasallowingtheoverheadcoststobedifferent.COST=b1+dOCC+b2N+lNOCC+uRegularschoolCOST=b1+b2N+u(OCC=NOCC=0))OccupationalschoolCOST=((b1+d)+((b2+l)N+u(OCC=1;;NOCC=N)SIB--BFSU,ECONOMETRICS63LECTURE10COSTNb1+db1OccupationalRegulardlSLOPEDUMMYVARIABLESThediagramillustratesthemodelgraphically..SIB--BFSU,ECONOMETRICS64LECTURE10SLOPEDUMMYVARIABLESHerearethedataforthefirsttenschools.NotetheweirdwayinwhichNOCCisdefined.SchoolTypeCOSTNOCCNOCC1 Occupational345,,000 62316232 Occupational537,00065316533 Regular170,000400 004 Occupational526.00066316635 Regular100,000563006 Regular28,000236007 Regular160,000307 008 Occupational45,,000 17311739 Occupational120,000146114610Occupational61,00099199SIB--BFSU,ECONOMETRICS65LECTURE10.DependentVariable::COSTMethod::LeastSquaresDate:05/16/04Time::21:17Sample::174Includedobservations::74VariableCoefficientStd..Error t--Statistic Prob..C 51475.25 31314.84 1..643797 0..1047N 152.2982 60.01932 2..537487 0..0134OCC--3501..177 41085.46 -0.0