第十講-虛擬變量DUMMY-VARIALBE課件

第十講虛擬變量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=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1'+b2N+uDUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESAnotherwayofhandlingthedifferencewouldbetohypothesizethatthecostfunctionforoccupationalschoolshasaninterceptb1'thatisgreaterthanthatforregularschools.b1b1'SIB-BFSU,ECONOMETRICS7LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESEffectively,wearehypothesizingthattheannualoverheadcostisdifferentforthetwotypesofschool,butthemarginalcostisthesame.Themarginalcostassumptionisnotveryplausibleandwewillrelaxitinduecourse.OCC=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1'+b2N+ub1b1'SIB-BFSU,ECONOMETRICS8LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESdLetusdefinedtobethedifferenceintheintercepts:d=b1'-b1.OCC=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1'+b2N+ub1b1'SIB-BFSU,ECONOMETRICS9LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThenb1'=b1+dandwecanrewritethecostfunctionforoccupationalschoolsasshown.b1+ddOCC=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1

+d+b2N+ub1SIB-BFSU,ECONOMETRICS10LECTURE10Combinedequation COST=b1+dOCC+b2N+uOCC=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1+d+b2N+uDUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWecannowcombinethetwocostfunctionsbydefiningadummyvariableOCCthathasvalue0forregularschoolsand1foroccupationalschools.db1b1+dSIB-BFSU,ECONOMETRICS11LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESDummyvariablesalwayshavetwovalues,0or1.IfOCCisequalto0,thecostfunctionbecomesthatforregularschools.IfOCCisequalto1,thecostfunctionbecomesthatforoccupationalschools.db1b1+dCombinedequation COST=b1+dOCC+b2N+uOCC=0Regularschool COST=b1+b2N+uOCC=1Occupationalschool COST=b1+d+b2N+uSIB-BFSU,ECONOMETRICS12LECTURE10DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWewillnowfitafunctionofthistypeusingactualdataforasampleof74secondaryschoolsinShanghai.SIB-BFSU,ECONOMETRICS13LECTURE10

School Type COST

N

OCC 1 Occupational 345,000 623 1 2 Occupational 537,000 653 1 3 Regular 170,000 400 0 4 Occupational 526.000 663 1 5 Regular 100,000 563 0 6 Regular 28,000 236 0 7 Regular 160,000 307 0 8 Occupational 45,000 173 1 9 Occupational 120,000 146 1 10 Occupational 61,000 99 1DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESThetableshowsthedataforthefirst10schoolsinthesample.Theannualcostismeasuredinyuan,oneyuanbeingworthabout20centsU.S.atthetime.Nisthenumberofstudentsintheschool.SIB-BFSU,ECONOMETRICS14LECTURE10

School Type COST

N

OCC 1 Occupational 345,000 623 1 2 Occupational 537,000 653 1 3 Regular 170,000 400 0 4 Occupational 526.000 663 1 5 Regular 100,000 563 0 6 Regular 28,000 236 0 7 Regular 160,000 307 0 8 Occupational 45,000 173 1 9 Occupational 120,000 146 1 10 Occupational 61,000 99 1DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESOCCisthedummyvariableforthetypeofschool.SIB-BFSU,ECONOMETRICS15LECTURE10.DependentVariable:COST Method:LeastSquares Date:05/16/04Time:19:22 Sample:174 Includedobservations:74 Variable Coefficient Std.Error t-Statistic Prob. C -33612.55 23573.47 -1.425864 0.1583 N 331.4493 39.75844 8.336578 0.0000 OCC 133259.1 20827.59 6.398201 0.0000 R-squared 0.615637 Meandependentvar 187418.0 AdjustedR-squared 0.604810 S.D.dependentvar 141969.9 S.E.ofregression 89248.09 Akaikeinfocriterion 25.67592 Sumsquaredresid 5.66E+11 Schwarzcriterion 25.76933 Loglikelihood -947.0092 F-statistic 56.86072 Durbin-Watsonstat 2.422989 Prob(F-statistic) 0.000000 DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWenowruntheregressionofCOSTonNandOCC,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:COST Method:LeastSquares Date:05/16/04Time:19:22 Sample:174 Includedobservations:74 Variable Coefficient Std.Error t-Statistic Prob. C -33612.55 23573.47 -1.425864 0.1583 N 331.4493 39.75844 8.336578 0.0000 OCC 133259.1 20827.59 6.398201 0.0000 R-squared 0.615637 Meandependentvar 187418.0 AdjustedR-squared 0.604810 S.D.dependentvar 141969.9 S.E.ofregression 89248.09 Akaikeinfocriterion 25.67592 Sumsquaredresid 5.66E+11 Schwarzcriterion 25.76933 Loglikelihood -947.0092 F-statistic 56.86072 Durbin-Watsonstat 2.422989 Prob(F-statistic) 0.000000 DUMMYVARIABLECLASSIFICATIONWITHTWOCATEGORIESWewillperformattestonthecoefficientofthedummyvariable.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+uGeneralSchool

COST=b1

+b2N+u(TECH=WORKER=VOC=0)SIB-BFSU,ECONOMETRICS35LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESIfanobservationrelatestoatechnicalschool,TECHwillbeequalto1andtheotherdummyvariableswillbe0.Theregressionmodelsimplifiesasshown.

COST=b1

+dTTECH+dWWORKER+dVVOC

+b2N+uGeneralSchool

COST=b1

+b2N+u(TECH=WORKER=VOC=0)TechnicalSchool

COST=(b1

+dT)+b2N+u(TECH=1;WORKER=VOC=0)SIB-BFSU,ECONOMETRICS36LECTURE10

COST=b1

+dTTECH+dWWORKER+dVVOC

+b2N+uGeneralSchool

COST=b1

+b2N+u(TECH=WORKER=VOC=0)TechnicalSchool

COST=(b1

+dT)+b2N+u(TECH=1;WORKER=VOC=0)SkilledWorkers’School

COST=(b1

+dW)+b2N+u(WORKER=1;TECH=VOC=0)VocationalSchool

COST=(b1

+dV)+b2N+u(VOC=1;TECH=WORKER=0)DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESTheregressionmodelsimplifiesinasimilarmannerinthecaseofobservationsrelatingtoskilledworkers’schoolsandvocationalschools.SIB-BFSU,ECONOMETRICS37LECTURE10COSTNb1+dTb1+dWb1+dVb1Workers’VocationaldWdVdTDUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThediagramillustratesthemodelgraphically.The

dcoefficientsaretheextraoverheadcostsofrunningtechnical,skilledworkers’,andvocationalschools,relativetotheoverheadcostofgeneralschools.TechnicalGeneralSIB-BFSU,ECONOMETRICS38LECTURE10COSTNdWdVdTNotethatwedonotmakeanypriorassumptionaboutthesize,oreventhesign,ofthedcoefficients.Theywillbeestimatedfromthesampledata.DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESWorkers’VocationalTechnicalGeneralb1+dTb1+dWb1+dVb1SIB-BFSU,ECONOMETRICS39LECTURE10

School Type COST N TECH WORKER VOC 1 Technical 345,000 623 1 0 0 2 Technical 537,000 653 1 0 0 3 General 170,000 400 0 0 0 4 Workers’ 526.000 663 0 1 0 5 General 100,000 563 0 0 0 6 Vocational 28,000 236 0 0 1 7 Vocational 160,000 307 0 0 1 8 Technical 45,000 173 1 0 0 9 Technical 120,000 146 1 0 0 10 Workers’ 61,000 99 0 1 0Herearethedataforthefirst10ofthe74schools.NotehowthevaluesofthedummyvariablesTECH,WORKER,andVOCaredeterminedbythetypeofschoolineachobservation.DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESSIB-BFSU,ECONOMETRICS40LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThescatterdiagramshowsthedatafortheentiresample,differentiatingbytypeofschool.SIB-BFSU,ECONOMETRICS41LECTURE10DependentVariable:COST Method:LeastSquares Date:05/16/04Time:20:32 Sample:174 Includedobservations:74 Variable Coefficient Std.Error t-Statistic Prob. C -54893.09 26673.08 -2.057996 0.0434 N 342.6335 40.21950 8.519090 0.0000 TECH 154110.9 26760.41 5.758915 0.0000 WORKER 143362.4 27852.80 5.147144 0.0000 VOC 53228.64 31061.65 1.713645 0.0911 R-squared 0.632050 Meandependentvar 187418.0 AdjustedR-squared 0.610719 S.D.dependentvar 141969.9 S.E.ofregression 88578.37 Akaikeinfocriterion 25.68634 Sumsquaredresid 5.41E+11 Schwarzcriterion 25.84202 Loglikelihood -945.3946 F-statistic 29.63132 Durbin-Watsonstat 2.503728 Prob(F-statistic) 0.000000 DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThecoefficientofNindicatesthatthemarginalcostperstudentperyearis343yuan.SIB-BFSU,ECONOMETRICS42LECTURE10DependentVariable:COST Method:LeastSquares Date:05/16/04Time:20:32 Sample:174 Includedobservations:74 Variable Coefficient Std.Error t-Statistic Prob. C -54893.09 26673.08 -2.057996 0.0434 N 342.6335 40.21950 8.519090 0.0000 TECH 154110.9 26760.41 5.758915 0.0000 WORKER 143362.4 27852.80 5.147144 0.0000 VOC 53228.64 31061.65 1.713645 0.0911 R-squared 0.632050 Meandependentvar 187418.0 AdjustedR-squared 0.610719 S.D.dependentvar 141969.9 S.E.ofregression 88578.37 Akaikeinfocriterion 25.68634 Sumsquaredresid 5.41E+11 Schwarzcriterion 25.84202 Loglikelihood -945.3946 F-statistic 29.63132 Durbin-Watsonstat 2.503728 Prob(F-statistic) 0.000000 DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThecoefficientsofTECH,WORKER,andVOCare154,000,143,000,and53,000,respectively,andshouldbeinterpretedastheadditionalannualoverheadcosts,relativetothoseofgeneralschools.SIB-BFSU,ECONOMETRICS43LECTURE10

COST=

-55,000+154,000TECH+143,000WORKER+53,000VOC+343NGeneralSchool

COST =-55,000+343N(TECH=WORKER=VOC=0)TechnicalSchool

COST =-55,000+154,000+343N(TECH=1;WORKER=VOC=0) =99,000+343NSkilledWorkers’School

COST =-55,000+143,000+343N(WORKER=1;TECH=VOC=0) =88,000+343NVocationalSchool

COST =-55,000+53,000+343N(VOC=1;TECH=WORKER=0) =-2,000+343NDUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESNotethatineachcasetheannualmarginalcostperstudentisestimatedat343yuan.Themodelspecificationassumesthatthisfiguredoesnotdifferaccordingtotypeofschool.^^^^^SIB-BFSU,ECONOMETRICS44LECTURE10DUMMYCLASSIFICATIONWITHMORETHANTWOCATEGORIESThefourcostfunctionsareillustratedgraphically.SIB-BFSU,ECONOMETRICS45LECTURE10TWOSETSOFDUMMYVARIABLESTheexplanatoryvariablesinaregressionmodelmayincludemultiplesetsofdummyvariables.Thissequenceprovidesanexampleofamodelwithtwotypes.

COST=b1

+d

OCC+e

RES+b2N+uSIB-BFSU,ECONOMETRICS46LECTURE10TWOSETSOFDUMMYVARIABLESWewillcontinuewiththeschoolcostfunctionmodelandextendittotakeaccountofthefactthatsomeoftheschoolsareresidential.

COST=b1

+d

OCC+e

RES+b2N+uSIB-BFSU,ECONOMETRICS47LECTURE10TWOSETSOFDUMMYVARIABLESTomodelthehigheroverheadcostsofresidentialschools,weintroduceadummyvariableRESwhichisequalto1forthemand0fornon-residentialschools.eistheextraannualoverheadcostofaresidentialschool,relativetothatofanon-residentialone.

COST=b1

+d

OCC+e

RES+b2N+uSIB-BFSU,ECONOMETRICS48LECTURE10TWOSETSOFDUMMYVARIABLESWewillalsomakeadistinctionbetweenoccupationalandregularschools,usingthedummyvariableOCCdefinedinthefirstsequence.

COST=b1

+d

OCC+e

RES+b2N+uSIB-BFSU,ECONOMETRICS49LECTURE10TWOSETSOFDUMMYVARIABLESInthecaseofanon-residentialoccupationalschool,RESis0andOCCis1,sotheoverheadcostincreasesbyd.Iftheschoolisbothoccupationalandresidential,itincreasesby(d+e).

COST=b1

+d

OCC+e

RES+b2N+uRegular,non-residential

COST=b1

+b2N+u(OCC=RES=0)Regular,residential

COST=(b1

+e

)+b2N+u(OCC=0;RES=1)Occupational,non-residential

COST=(b1

+d

)+b2N+u(OCC=1;RES=0)Occupational,residential

COST=(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.

School TypeResidential? COST

N

OCC RES 1 Occupational No 345,000 623 1 0 2 Occupational Yes 537,000 653 1 1 3 Regular No 170,000 400 0 0 4 Occupational Yes 526.000 663 1 1 5 Regular No 100,000 563 0 0 6 Regular No 28,000 236 0 0 7 Regular Yes 160,000 307 0 1 8 Occupational No 45,000 173 1 0 9 Occupational No 120,000 146 1 0 10 Occupational No 61,000 99 1 0SIB-BFSU,ECONOMETRICS52LECTURE10DependentVariable:COST Method:LeastSquares Date:05/16/04Time:21:06 Sample:174 Includedobservations:74 Variable Coefficient Std.Error t-Statistic Prob. C -29045.27 23291.54 -1.247031 0.2165 N 321.8330 39.40225 8.167884 0.0000 OCC 109564.6 24039.58 4.557674 0.0000 RES 57909.01 30821.31 1.878863 0.0644 R-squared 0.634090 Meandependentvar 187418.0 AdjustedR-squared 0.618409 S.D.dependentvar 141969.9 S.E.ofregression 87699.16 Akaikeinfocriterion 25.65375 Sumsquaredresid 5.38E+11 Schwarzcriterion 25.77829 Loglikelihood -945.1888 F-statistic 40.43470 Durbin-Watsonstat 2.443610 Prob(F-statistic) 0.000000 TWOSETSOFDUMMYVARIABLESWewillstartbyinterpretingtheregressioncoefficients.ThecoefficientofNindicatesthatthemarginalcostperstudentis322yuanperyear.SIB-BFSU,ECONOMETRICS53LECTURE10TWOSETSOFDUMMYVARIABLES

COST=

-29,000+110,000OCC+58,000RES+322NRegular,non-residential

COST =-29,000+322N(OCC=RES=0)Regular,residential

COST =-29,000+58,000+322N(OCC=0;RES=1) =29,000+322NOccupational,non-residential

COST =-29,000+110,000+322N(OCC=1;RES=0) =81,000+322NOccupational,residential

COST =-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

+d

OCC+b2N+lNOCC

+u

SIB-BFSU,ECONOMETRICS60LECTURE10SLOPEDUMMYVARIABLESInthecaseofaregularschool,OCCis0andhencesoalsoisNOCC.Themodelreducestoitsbasiccomponents.

COST=b1

+d

OCC+b2N+lNOCC

+u Regularschool

COST=b1

+b2N+u (OCC=NOCC=0)

SIB-BFSU,ECONOMETRICS61LECTURE10SLOPEDUMMYVARIABLESInthecaseofanoccupationalschool,OCCisequalto1andNOCCisequaltoN.Theequationsimplifiesasshown.

COST=b1

+d

OCC+b2N+lNOCC

+u Regularschool

COST=b1

+b2N+u (OCC=NOCC=0) Occupationalschool

COST=(b1

+d

)+(b2

+l)N+u (OCC=1;NOCC=N)SIB-BFSU,ECONOMETRICS62LECTURE10SLOPEDUMMYVARIABLESThemodelnowallowsthemarginalcostperstudenttobeanamountlgreaterthanthatinregularschools,aswellasallowingtheoverheadcoststobedifferent.

COST=b1

+d

OCC+b2N+lNOCC

+u Regularschool

COST=b1

+b2N+u (OCC=NOCC=0) Occupationalschool

COST=(b1

+d

)+(b2

+l)N+u (OCC=1;NOCC=N)SIB-BFSU,ECONOMETRICS63LECTURE10COSTNb1+db1OccupationalRegulardlSLOPEDUMMYVARIABLESThediagramillustratesthemodelgraphically.SIB-BFSU,ECONOMETRICS64LECTURE10SLOPEDUMMYVARIABLESHerearethedataforthefirsttenschools.NotetheweirdwayinwhichNOCCisdefined.

School Type COST

N

OCC NOCC 1 Occupational 345,000 623 1 623 2 Occupational 537,000 653 1 653 3 Regular 170,000 400 0 0 4 Occupational 526.000 663 1 663 5 Regular 100,000 563 0 0 6 Regular 28,000 236 0 0 7 Regular 160,000 307 0 0 8 Occupational 45,000 173 1 173 9 Occupational 120,000 146 1 146 10 Occupational 61,000 99 1 99SIB-BFSU,ECONOMETRICS65LECTURE10.DependentVariable:COST Method:LeastSquares Date:05/16/04Time:21:17 Sample:174 Includedobservations:74 Vari