1 two sets of dummy variables the explanatory variables in a regression model may include multiple...
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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 + 2N + u
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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 + 2N + u
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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 + 2N + u
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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 + 2N + u
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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 + 2N + u
COST = 1+ 2N + u
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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 + 2N + u
COST = 1+ 2N + u
COST = (1+ ) + 2N + uRegular, residential(OCC = 0; RES = 1)
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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 + 2N + u
COST = 1+ 2N + u
COST = (1+ ) + 2N + uRegular, residential(OCC = 0; RES = 1)
COST = (1+ + ) + 2N + u
Occupational, residential(OCC = 1; RES = 1)
COST = (1+ ) + 2N + uOccupational, nonresidential (OCC = 1; RES = 0)
N
1++1+
1+1
Occupational, residential
Regular, nonresidential
+
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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
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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
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.
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School Type Residential? 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 0
TWO SETS OF DUMMY VARIABLES
Here is a scatter diagram showing the four types of school.
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TWO SETS OF DUMMY VARIABLES
N
COST
-100000
0
100000
200000
300000
400000
500000
600000
0 200 400 600 800 1000 1200
Nonresidential regular Residential regular
Nonresidential occupational Residential occupational
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
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.
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TWO SETS OF DUMMY VARIABLES
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
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.
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TWO SETS OF DUMMY VARIABLES
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
The coefficient of OCC indicates that the annual overhead costs of occupational schools are 110,000 yuan more than those of regular schools.
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TWO SETS OF DUMMY VARIABLES
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
The coefficient of RES indicates that the annual overhead costs of residential schools are 58,000 yuan greater than those of nonresidential schools.
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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.
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TWO SETS OF DUMMY VARIABLES
Regular, nonresidential(OCC = RES = 0)
COST = –29,000 + 110,000OCC + 58,000RES + 322N^
^COST = –29,000 + 322N
Putting RES equal to 1, but keeping OCC at 0, we obtain the cost function for residential regular schools.
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TWO SETS OF DUMMY VARIABLES
Regular, nonresidential(OCC = RES = 0)
COST = –29,000 + 110,000OCC + 58,000RES + 322N
Regular, residential(OCC = 0; RES = 1)
^
^COST = –29,000 + 322N
^COST = –29,000 + 58,000 + 322N= 29,000 + 322N
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.
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TWO SETS OF DUMMY VARIABLES
Regular, nonresidential(OCC = RES = 0)
COST = –29,000 + 110,000OCC + 58,000RES + 322N
Regular, residential(OCC = 0; RES = 1)
Occupational, residential(OCC = 1; RES = 1)
Occupational, nonresidential(OCC = 1; RES = 0)
^
^COST = –29,000 + 322N
^COST = –29,000 + 58,000 + 322N= 29,000 + 322N
^COST = –29,000 + 110,000 + 322N= 81,000 + 322N
^COST = –29,000 + 110,000 + 58,000 + 322N= 139,000 + 322N
Here is the scatter diagram with the four cost functions implicit in the regression result.
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TWO SETS OF DUMMY VARIABLES
-100000
0
100000
200000
300000
400000
500000
600000
0 200 400 600 800 1000 1200
Nonresidential regular Residential regular
Nonresidential occupational Residential occupational
O, R
O, NR, R
R, N
N
COST
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
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.
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TWO SETS OF DUMMY VARIABLES
. reg COST N OCC RES
Source | SS df MS Number of obs = 74---------+------------------------------ F( 3, 70) = 40.43 Model | 9.3297e+11 3 3.1099e+11 Prob > F = 0.0000Residual | 5.3838e+11 70 7.6911e+09 R-squared = 0.6341---------+------------------------------ Adj R-squared = 0.6184 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 87699
------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 321.833 39.40225 8.168 0.000 243.2477 400.4183 OCC | 109564.6 24039.58 4.558 0.000 61619.15 157510 RES | 57909.01 30821.31 1.879 0.064 -3562.137 119380.2 _cons | -29045.27 23291.54 -1.247 0.217 -75498.78 17408.25------------------------------------------------------------------------------
However, the t ratio for the coefficient of RES is only 1.87. 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).
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TWO SETS OF DUMMY VARIABLES
Copyright Christopher Dougherty 2012.
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2012.11.04