第十讲 虚拟变量DUMMY VARIALBE

第十讲 虚拟变量DUMMY VARIALBE DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES TWO SETS OF DUMMY VARIABLES SLOPE DUMMY VARIABLESSIB-BFSU, ECONOMETRICS1DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESThis sequence explains how you can include qualitative explanatory variables in your regression model.NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS2DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESSuppose that you have data on the annual recurrent expenditure, COST, and the number of students enrolled, N, for a sample of secondary schools, of which there are two types: regular and occupational.NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS3DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESThe occupational schools aim to provide skills for specific occupations and they tend to be relatively expensive to run because they need to maintain specialized workshops.NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS4DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESOne way of dealing with the difference in the costs would be to run separate regressions for the two types of school.NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS5DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESHowever this would have the drawback that you would be running regressions with two small samples instead of one large one, with an adverse effect on the precision of the estimates of the coefficients.NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS6OCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + b b2N + uDUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESAnother way of handling the difference would be to hypothesize that the cost function for occupational schools has an intercept b b1 that is greater than that for regular schools.NCOSTOccupational schoolsRegular schoolsb b1b b1SIB-BFSU, ECONOMETRICS7DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESEffectively, we are hypothesizing that the annual overhead cost is different for the two types of school, but the marginal cost is the same. The marginal cost assumption is not very plausible and we will relax it in due course.NCOSTOccupational schoolsRegular schoolsOCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + b b2N + ub b1b b1SIB-BFSU, ECONOMETRICS8DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESd dLet us define d d to be the difference in the intercepts: d d = b b1 - b b1.NCOSTOccupational schoolsRegular schoolsOCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + b b2N + ub b1b b1SIB-BFSU, ECONOMETRICS9DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESThen b b1 = b b1 + d d and we can rewrite the cost function for occupational schools as shown.NCOSTOccupational schoolsRegular schoolsb b1+d dd dOCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + d d + b b2N + ub b1SIB-BFSU, ECONOMETRICS10Combined equationCOST = b b1 + d d OCC + b b2N + uOCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + d d + b b2N + uDUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESWe can now combine the two cost functions by defining a dummy variable OCC that has value 0 for regular schools and 1 for occupational schools.NCOSTOccupational schoolsRegular schoolsd db b1b b1+d dSIB-BFSU, ECONOMETRICS11DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESDummy variables always have two values, 0 or 1. If OCC is equal to 0, the cost function becomes that for regular schools. If OCC is equal to 1, the cost function becomes that for occupational schools.NCOSTOccupational schoolsRegular schoolsd db b1b b1+d dCombined equationCOST = b b1 + d d OCC + b b2N + uOCC = 0 Regular schoolCOST = b b1 + b b2N + uOCC = 1 Occupational schoolCOST = b b1 + d d + b b2N + uSIB-BFSU, ECONOMETRICS12DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESWe will now fit a function of this type using actual data for a sample of 74 secondary schools in Shanghai.01000002000003000004000005000006000007000000200400600800100012001400NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS13School TypeCOST N OCC1Occupational345,00062312Occupational 537,00065313Regular 170,00040004Occupational 526.00066315Regular100,00056306Regular 28,00023607Regular 160,00030708Occupational 45,00017319Occupational 120,000146110 Occupational61,000991DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESThe table shows the data for the first 10 schools in the sample. The annual cost is measured in yuan, one yuan being worth about 20 cents U.S. at the time. N is the number of students in the school. SIB-BFSU, ECONOMETRICS14School TypeCOST N OCC1Occupational345,00062312Occupational 537,00065313Regular 170,00040004Occupational 526.00066315Regular100,00056306Regular 28,00023607Regular 160,00030708Occupational 45,00017319Occupational 120,000146110 Occupational61,000991DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESOCC is the dummy variable for the type of school.SIB-BFSU, ECONOMETRICS15. Dependent Variable: COSTMethod: Least SquaresDate: 05/16/04 Time: 19:22Sample: 1 74Included observations: 74VariableCoefficientStd. Errort-StatisticProb. C-33612.5523573.47-1.4258640.1583N331.449339.758448.3365780.0000OCC133259.120827.596.3982010.0000R-squared0.615637 Mean dependent var187418.0Adjusted R-squared0.604810 S.D. dependent var141969.9S.E. of regression89248.09 Akaike info criterion25.67592Sum squared resid5.66E+11 Schwarz criterion25.76933Log likelihood-947.0092 F-statistic56.86072Durbin-Watson stat2.422989 Prob(F-statistic)0.000000DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESWe now run the regression of COST on N and OCC, treating OCC just like any other explanatory variable, despite its artificial nature. SIB-BFSU, ECONOMETRICS16 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESCOST = -34,000 + 133,000OCC + 331NThe regression results have been rewritten in equation form. From it we can derive cost functions for the two types of school by setting OCC equal to 0 or 1.SIB-BFSU, ECONOMETRICS17 Regular School (OCC = 0)DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESCOST = -34,000 + 133,000OCC + 331NCOST = -34,000 + 331NIf OCC is equal to 0, we get the equation for regular schools, as shown. It implies that the marginal cost per student per year is 331 yuan and that the annual overhead cost is -34,000 yuan.SIB-BFSU, ECONOMETRICS18 Regular School (OCC = 0)DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESCOST = -34,000 + 133,000OCC + 331NCOST = -34,000 + 331NObviously having a negative intercept does not make any sense at all and it suggests that the model is misspecified in some way. We will come back to this later.SIB-BFSU, ECONOMETRICS19 Regular School (OCC = 0) Occupational School (OCC = 1)DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESPutting OCC equal to 1, we estimate the annual overhead cost of an occupational school to be 99,000 yuan. The marginal cost is the same as for regular schools. It must be, given the model specification.COST = -34,000 + 133,000OCC + 331NCOST = -34,000 + 331NCOST = -34,000 + 133,000 + 331N= 99,000 + 331NSIB-BFSU, ECONOMETRICS20DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESThe scatter diagram shows the data and the two cost functions derived from the regression results.-10000001000002000003000004000005000006000007000000200400600800100012001400NCOSTOccupational schoolsRegular schoolsSIB-BFSU, ECONOMETRICS21. Dependent Variable: COSTMethod: Least SquaresDate: 05/16/04 Time: 19:22Sample: 1 74Included observations: 74VariableCoefficientStd. Errort-StatisticProb. C-33612.5523573.47-1.4258640.1583N331.449339.758448.3365780.0000OCC133259.120827.596.3982010.0000R-squared0.615637 Mean dependent var187418.0Adjusted R-squared0.604810 S.D. dependent var141969.9S.E. of regression89248.09 Akaike info criterion25.67592Sum squared resid5.66E+11 Schwarz criterion25.76933Log likelihood-947.0092 F-statistic56.86072Durbin-Watson stat2.422989 Prob(F-statistic)0.000000DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIESWe will perform a t test on the coefficient of the dummy variable. our null hypothesis is that there is no difference in the overhead costs of the two types of school. The t statistic is 6.40, so it is rejected at the 0.1% significance level.SIB-BFSU, ECONOMETRICS22DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThis sequence explains how to extend the dummy variable technique to handle a qualitative explanatory variable which has more than two categories.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS23DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIn the previous sequence we used a dummy variable to differentiate between regular and occupational schools when fitting a cost function.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS24DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIn actual fact there are two types of regular secondary school in Shanghai. There are general schools, which provide the usual academic education, and vocational schools.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS25DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESAs their name implies, the vocational schools are meant to impart occupational skills as well as give an academic education.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS26DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESHowever the vocational component of the curriculum is typically quite small and the schools are similar to the general schools. Often they are just general schools with a couple of workshops added.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS27DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESLikewise there are two types of occupational school. There are technical schools training technicians and skilled workers schools training craftsmen.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS28DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESSo now the qualitative variable has four categories. The standard procedure is to choose one category as the reference category and to define dummy variables for each of the others.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS29DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIn general it is good practice to select the most normal or basic category as the reference category, if one category is in some sense more normal or basic than the others.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS30DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIn the Shanghai sample it is sensible to choose the general schools as the reference category. They are the most numerous and the other schools are variations of them.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS31DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESAccordingly we will define dummy variables for the other three types. TECH will be the dummy for the technical schools: TECH is equal to 1 if the observation relates to a technical school, 0 otherwise.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS32DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESEach of the dummy variables will have a coefficient which represents the extra overhead costs of the schools, relative to the reference category.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS33DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESNote that you do not include a dummy variable for the reference category, and that is the reason that the reference category is usually described as the omitted category.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uSIB-BFSU, ECONOMETRICS34DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIf an observation relates to a general school, the dummy variables are all 0 and the regression model is reduced to its basic components.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uGeneral SchoolCOST = b b1 + b b2N + u(TECH = WORKER = VOC = 0)SIB-BFSU, ECONOMETRICS35DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESIf an observation relates to a technical school, TECH will be equal to 1 and the other dummy variables will be 0. The regression model simplifies as shown.COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uGeneral SchoolCOST = b b1 + b b2N + u(TECH = WORKER = VOC = 0)Technical SchoolCOST = (b b1 + d dT) + b b2N + u(TECH = 1; WORKER = VOC = 0)SIB-BFSU, ECONOMETRICS36COST = b b1 + d dTTECH + d dWWORKER + d dVVOC + b b2N + uGeneral SchoolCOST = b b1 + b b2N + u(TECH = WORKER = VOC = 0)Technical SchoolCOST = (b b1 + d dT) + b b2N + u(TECH = 1; WORKER = VOC = 0)Skilled Workers SchoolCOST = (b b1 + d dW) + b b2N + u(WORKER = 1; TECH = VOC = 0)Vocational SchoolCOST = (b b1 + d dV) + b b2N + u(VOC = 1; TECH = WORKER = 0)DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe regression model simplifies in a similar manner in the case of observations relating to skilled workers schools and vocational schools.SIB-BFSU, ECONOMETRICS37COSTNb b1+d dTb b1+d dWb b1+d dVb b1WorkersVocationald dWd dVd dTDUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe diagram illustrates the model graphically. The d d coefficients are the extra overhead costs of running technical, skilled workers, and vocational schools, relative to the overhead cost of general schools.TechnicalGeneralSIB-BFSU, ECONOMETRICS38COSTNd dWd dVd dTNote that we do not make any prior assumption about the size, or even the sign, of the d d coefficients. They will be estimated from the sample data.DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESWorkersVocationalTechnicalGeneralb b1+d dTb b1+d dWb b1+d dVb b1SIB-BFSU, ECONOMETRICS39School TypeCOST N TECH WORKERVOC1Technical345,0006231002Technical 537,0006531003General 170,0004000004Workers 526.0006630105General 100,0005630006Vocational 28,0002360017Vocational 160,0003070018Technical 45,0001731009Technical 120,00014610010 Workers 61,00099010Here are the data for the first 10 of the 74 schools. Note how the values of the dummy variables TECH, WORKER, and VOC are determined by the type of school in each observation.DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESSIB-BFSU, ECONOMETRICS40DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe scatter diagram shows the data for the entire sample, differentiating by type of school.01000002000003000004000005000006000007000000200400600800100012001400NCOSTTechnical schoolsVocational schoolsGeneral schoolsWorkers schoolsSIB-BFSU, ECONOMETRICS41Dependent Variable: COSTMethod: Least SquaresDate: 05/16/04 Time: 20:32Sample: 1 74Included observations: 74VariableCoefficientStd. Errort-StatisticProb. C-54893.0926673.08-2.0579960.0434N342.633540.219508.5190900.0000TECH154110.926760.415.7589150.0000WORKER143362.427852.805.1471440.0000VOC53228.6431061.651.7136450.0911R-squared0.632050 Mean dependent var187418.0Adjusted R-squared0.610719 S.D. dependent var141969.9S.E. of regression88578.37 Akaike info criterion25.68634Sum squared resid5.41E+11 Schwarz criterion25.84202Log likelihood-945.3946 F-statistic29.63132Durbin-Watson stat2.503728 Prob(F-statistic)0.000000DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe coefficient of N indicates that the marginal cost per student per year is 343 yuan.SIB-BFSU, ECONOMETRICS42Dependent Variable: COSTMethod: Least SquaresDate: 05/16/04 Time: 20:32Sample: 1 74Included observations: 74VariableCoefficientStd. Errort-StatisticProb. C-54893.0926673.08-2.0579960.0434N342.633540.219508.5190900.0000TECH154110.926760.415.7589150.0000WORKER143362.427852.805.1471440.0000VOC53228.6431061.651.7136450.0911R-squared0.632050 Mean dependent var187418.0Adjusted R-squared0.610719 S.D. dependent var141969.9S.E. of regression88578.37 Akaike info criterion25.68634Sum squared resid5.41E+11 Schwarz criterion25.84202Log likelihood-945.3946 F-statistic29.63132Durbin-Watson stat2.503728 Prob(F-statistic)0.000000DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe coefficients of TECH, WORKER, and VOC are 154,000, 143,000, and 53,000, respectively, and should be interpreted as the additional annual overhead costs, relative to those of general schools.SIB-BFSU, ECONOMETRICS43COST = -55,000 + 154,000TECH + 143,000WORKER + 53,000VOC + 343NGeneral SchoolCOST = -55,000 + 343N(TECH = WORKER = VOC = 0)Technical SchoolCOST = -55,000 + 154,000 + 343N(TECH = 1; WORKER = VOC = 0)= 99,000 + 343NSkilled Workers SchoolCOST = -55,000 + 143,000 + 343N(WORKER = 1; TECH = VOC = 0)= 88,000 + 343NVocational SchoolCOST = -55,000 + 53,000 + 343N(VOC = 1; TECH = WORKER = 0)= -2,000 + 343NDUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESNote that in each case the annual marginal cost per student is estimated at 343 yuan. The model specification assumes that this figure does not differ according to type of school.SIB-BFSU, ECONOMETRICS44DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIESThe four cost functions are illustrated graphically.-10000001000002000003000004000005000006000007000000200400600800100012001400NCOSTTechnical schoolsVocational schoolsGeneral schoolsWorkers schoolsSIB-BFSU, ECONOMETRICS45TWO SETS OF DUMMY VARIABLESThe explanatory variables in a regression model may include multiple sets of dummy variables. This sequence provides an example of a model with two types.COST = b b1 + d d OCC + e e RES + b b2N + uSIB-BFSU, ECONOMETRICS46TWO SETS OF DUMMY VARIABLESWe will continue with the school cost function model and extend it to take account of the fact that some of the schools are residential.COST = b b1 + d d OCC + e e RES + b b2N + uSIB-BFSU, ECONOMETRICS47TWO SETS OF DUMMY VARIABLESTo model the higher overhead costs of residential schools, we introduce a dummy variable RES which is equal to 1 for them and 0 for non-residential schools. e e is the extra annual overhead cost of a residential school, relative to that of a non-residential one.COST = b b1 + d d OCC + e e RES + b b2N + uSIB-BFSU, ECONOMETRICS48TWO SETS OF DUMMY VARIABLESWe will also make a distinction between occupational and regular schools, using the dummy variable OCC defined in the first sequence. COST = b b1 + d d OCC + e e RES + b b2N + uSIB-BFSU, ECONOMETRICS49TWO SETS OF DUMMY VARIABLESIn the case of a non-residential occupational school, RES is 0 and OCC is 1, so the overhead cost increases by d d. If the school is both occupational and residential, it increases by (d d + e e).COST = b b1 + d d OCC + e e RES + b b2N + uRegular, non-residentialCOST = b b1 + b b2N + u(OCC = RES = 0)Regular, residentialCOST = (b b1 + e e ) + b b2N + u(OCC = 0; RES = 1)Occupational, non-residentialCOST = (b b1 + d d ) + b b2N + u(OCC = 1; RES = 0)Occupational, residentialCOST = (b b1 + d d + e e ) + b b2N + u(OCC = RES = 1)SIB-BFSU, ECONOMETRICS50COSTNb b1+d d +e eb b1+d db b1+e eb b1Occupational, residentialRegular, non-residentiald de ed d +e ee eOccupational,non-residentialRegular,residentialTWO SETS OF DUMMY VARIABLESThe diagram illustrates the model graphically. Note that the effects of the different components of the model are assumed to be separate and additive in this specification.SIB-BFSU, ECONOMETRICS51TWO SETS OF DUMMY VARIABLESHere are the data for the first 10 schools. Note how the values of the dummy variables vary according to the characteristics of the school.School Type Residential?COST N OCCRES1OccupationalNo345,000623102Occupational Yes537,000653113Regular No170,000400004Occupational Yes526.000663115RegularNo100,000563006Regular No28,000236007Regular Yes160,000307018Occupational No45,000173109Occupational No120,0001461010 OccupationalNo61,0009910SIB-BFSU, ECONOMETRICS52Dependent Variable: COSTMethod: Least SquaresDate: 05/16/04 Time: 21:06Sample: 1 74Included observations: 74VariableCoefficientStd. Errort-StatisticProb. C-29045.2723291.54-1.2470310.2165N321.833039.402258.1678840.0000OCC109564.62