1 TWO SETS OF DUMMY VARIABLES The 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 =  1  +  OCC +  RES +  2 N + u

2 We will continue with the school cost function model and extend it to take account of the fact that some of the schools are residential. TWO SETS OF DUMMY VARIABLES COST =  1  +  OCC +  RES +  2 N + u

3 To model the higher overhead costs of residential schools, we introduce a dummy variable RES which is equal to 1 for them and 0 for nonresidential schools.  is the extra annual overhead cost of a residential school, relative to that of a nonresidential one. TWO SETS OF DUMMY VARIABLES COST =  1  +  OCC +  RES +  2 N + u

4 We will also make a distinction between occupational and regular schools, using the dummy variable OCC defined in the first sequence. (It would be better to use the four-category school classification, and in practice we would, but it would complicate the graphics.) TWO SETS OF DUMMY VARIABLES COST =  1  +  OCC +  RES +  2 N + u

5 If a school has a regular curriculum and is nonresidential, both dummy variables are 0 and the cost function simplifies to its basic components. TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST =  1  +  OCC +  RES +  2 N + u COST =  1  +  2 N + u

6 TWO SETS OF DUMMY VARIABLES For a residential regular school, RES is equal to 1 and the intercept increases by an amount . TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST =  1  +  OCC +  RES +  2 N + u COST =  1  +  2 N + u COST = (  1  +  ) +  2 N + u Regular, residential (OCC = 0; RES = 1)

7 In the case of a nonresidential occupational school, RES is 0 and OCC is 1, so the overhead cost increases by . If the school is both occupational and residential, it increases by (  +  ). TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST =  1  +  OCC +  RES +  2 N + u COST =  1  +  2 N + u COST = (  1  +  ) +  2 N + u Regular, residential (OCC = 0; RES = 1) COST = (  1  +  +  ) +  2 N + u Occupational, residential (OCC = 1; RES = 1) COST = (  1  +  ) +  2 N + u Occupational, nonresidential (OCC = 1; RES = 0)

N  1 +  +  1+1+ 1+1+ 11 Occupational, residential Regular, nonresidential    +  8  Occupational, nonresidential Regular, residential The 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. TWO SETS OF DUMMY VARIABLES COST

9 In particular, we are assuming that the extra overhead cost of a residential school is the same for regular and occupational schools. TWO SETS OF DUMMY VARIABLES N  1 +  +  1+1+ 1+1+ 11 Occupational, residential Regular, nonresidential    +   Occupational, nonresidential Regular, residential COST

Here are the data for the first 10 schools. Note how the values of the dummy variables vary according to the characteristics of the school. 10 School Type Residential?COST N OCCRES 1OccupationalNo345, Occupational Yes537, Regular No170, Occupational Yes RegularNo100, Regular No28, Regular Yes160, Occupational No45, Occupational No120, OccupationalNo61, TWO SETS OF DUMMY VARIABLES

Here is a scatter diagram showing the four types of school. 11 TWO SETS OF DUMMY VARIABLES N COST

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | Here is the Stata output for the regression. We will start by interpreting the regression coefficients. The coefficient of N indicates that the marginal cost per student is 322 yuan per year. 12 TWO SETS OF DUMMY VARIABLES

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | The constant provides an estimate of the annual overhead cost of the reference category, nonresidential regular schools. It is still negative, which does not make any sense. 13 TWO SETS OF DUMMY VARIABLES

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | The coefficient of OCC indicates that the annual overhead costs of occupational schools are 110,000 yuan more than those of regular schools. 14 TWO SETS OF DUMMY VARIABLES

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | The coefficient of RES indicates that the annual overhead costs of residential schools are 58,000 yuan greater than those of nonresidential schools. 15 TWO SETS OF DUMMY VARIABLES

The regression result is shown at the top in equation form. Putting both dummy variables equal to 0, we obtain the implicit cost function for nonresidential regular schools. 16 TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST = –29, ,000OCC + 58,000RES + 322N ^ ^ COST= –29, N

Putting RES equal to 1, but keeping OCC at 0, we obtain the cost function for residential regular schools. 17 TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST = –29, ,000OCC + 58,000RES + 322N Regular, residential (OCC = 0; RES = 1) ^ ^ COST= –29, N ^ COST= –29, , N = 29, N

And similarly the cost functions for nonresidential and residential occupational schools are derived by putting OCC equal to 1 and RES equal to 0 and 1, respectively. 18 TWO SETS OF DUMMY VARIABLES Regular, nonresidential (OCC = RES = 0) COST = –29, ,000OCC + 58,000RES + 322N Regular, residential (OCC = 0; RES = 1) Occupational, residential (OCC = 1; RES = 1) Occupational, nonresidential (OCC = 1; RES = 0) ^ ^ COST= –29, N ^ COST= –29, , N = 29, N ^ COST= –29, , N = 81, N ^ COST= –29, , , N = 139, N

Here is the scatter diagram with the four cost functions implicit in the regression result. 19 TWO SETS OF DUMMY VARIABLES N COST

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | t tests and F tests can be performed in the usual way. The coefficient of the occupational school dummy variable is significantly different from 0 at the 0.1% significance level. 20 TWO SETS OF DUMMY VARIABLES

. reg COST N OCC RES Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | RES | _cons | However, the t ratio for the coefficient of RES is only Fortunately we may perform a one-sided test (why?), so it is significantly different from 0 at the 5% level (but not the 1% level). 21 TWO SETS OF DUMMY VARIABLES

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 5.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics