THE EFFECTS OF CHANGING THE REFERENCE CATEGORY 1 In the previous sequence we chose general academic schools as the reference (omitted) category and defined

THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

1

In the previous sequence we chose general academic schools as the reference (omitted) category and defined dummy variables for the other categories.

N

COST

-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200

Technical schools Workers' schools Vocational schools General schools

2

This enabled us to compare the overhead costs of the other schools with those of general schools and to test whether the differences were significant.


N

COST

-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200


3

However, suppose that we were interested in testing whether the overhead costs of skilled workers’ schools were different from those of the other types of school. How could we do this?


N

COST

-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200


4

It is possible to perform a t test using the variance-covariance matrix of the regression coefficients to calculate the relevant standard errors. But it is a pain and it is easy to make arithmetical errors.


N

COST

-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200


5

It is much simpler to re-run the regression making skilled workers’ schools the reference category. Now we need to define a dummy variable GEN for the general schools.


N

COST

-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200


6

The model is shown in equation form. Note that there is no longer a dummy variable for skilled workers’ schools since they form the reference category.


COST = 1+ TTECH + VVOC + GGEN + 2N + u

7

In the case of observations relating to skilled workers’ schools, all the dummy variables are 0 and the model simplifies to the intercept and the term involving N.


Skilled workers' school(TECH = VOC = GEN = 0)


COST = 1+ 2N + u

8

In the case of observations relating to technical schools, TECH is equal to 1 and the intercept increases by an amount T.




COST = 1+ 2N + u

COST = (1+ T) + 2N + uTechnical school(TECH = 1; VOC = GEN = 0)

9

Note that T should now be interpreted as the extra overhead cost of a technical school relative to that of a skilled workers’ school.




COST = 1+ 2N + u


10

Similarly one can derive the implicit cost functions for vocational and general schools, their coefficients also being interpreted as their extra overhead costs relative to those of skilled workers’ schools.




COST = 1+ 2N + u


COST = (1+ G) + 2N + uGeneral school(GEN = 1; TECH = VOC = 0)

COST = (1+ V) + 2N + uVocational school(VOC = 1; TECH = GEN = 0)

11

This diagram illustrates the model graphically. Note that the shifts are measured from the line for skilled workers’ schools.

N

1+T

1+V

Technical

Workers’

Vocational

General

GV

T

1

1+G


COST

12

Here are the data for the first 10 of the 74 schools with skilled workers’ schools as the reference category.


School Type COST N TECH VOC GEN

1 Technical 345,000 623 1 0 0

2 Technical 537,000 653 1 0 0

3 General 170,000 400 0 0 1

4 Workers’ 526.000 663 0 0 0

5 General 100,000 563 0 0 1

6 Vocational 28,000 236 0 1 0

7 Vocational 160,000 307 0 1 0

8 Technical 45,000 173 1 0 0

9 Technical 120,000 146 1 0 0

10 Workers’ 61,000 99 0 0 0

. reg COST N TECH VOC GEN

Source | SS df MS Number of obs = 74---------+------------------------------ F( 4, 69) = 29.63 Model | 9.2996e+11 4 2.3249e+11 Prob > F = 0.0000Residual | 5.4138e+11 69 7.8461e+09 R-squared = 0.6320---------+------------------------------ Adj R-squared = 0.6107 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 88578

------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 342.6335 40.2195 8.519 0.000 262.3978 422.8692 TECH | 10748.51 30524.87 0.352 0.726 -50146.93 71643.95 VOC | -90133.74 33984.22 -2.652 0.010 -157930.4 -22337.07 GEN | -143362.4 27852.8 -5.147 0.000 -198927.2 -87797.57 _cons | 88469.29 28849.56 3.067 0.003 30916.01 146022.6------------------------------------------------------------------------------

13

Here is the Stata output for the regression. We will focus first on the regression coefficients.


14

The regression result is shown written as an equation.


COST = 88,000 + 11,000TECH – 90,000VOC – 143,000GEN + 343N^

15

Putting all the dummy variables equal to 0, we obtain the equation for the reference category, the skilled workers’ schools.



COST = 88,000 + 11,000TECH – 90,000VOC – 143,000GEN + 343N^

^COST = 88,000 + 343N

16

Putting TECH equal to 1 and VOC and GEN equal to 0, we obtain the equation for the technical schools.



COST = 88,000 + 11,000TECH – 90,000VOC – 143,000GEN + 343N

Technical school(TECH = 1; VOC = GEN = 0)

^

^COST = 88,000 + 343N

^COST = 88,000 + 11,000 + 343N= 99,000 + 343N

17

And similarly we obtain the equations for the vocational and general schools, putting VOC and GEN equal to 1 in turn.





General school(VOC = 1; TECH = WORKER = 0)

Vocational school(VOC = 1; TECH = GEN = 0)

^

^COST = 88,000 + 343N

^COST = 88,000 + 11,000 + 343N= 99,000 + 343N

^COST = 88,000 – 90,000 + 343N= –2,000 + 343N

^COST = 88,000 – 143,000 + 343N= –55,000 + 343N

18

Note that the cost functions turn out to be exactly the same as when we used general schools as the reference category.





General school(VOC = 1; TECH = WORKER = 0)

Vocational school(VOC = 1; TECH = GEN = 0)

^

^COST = 88,000 + 343N

^COST = 88,000 + 11,000 + 343N= 99,000 + 343N

^COST = 88,000 – 90,000 + 343N= –2,000 + 343N

^COST = 88,000 – 143,000 + 343N= –55,000 + 343N

19

Consequently the scatter diagram with regression lines is exactly the same as before.


-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200


N

COST

20

The goodness of fit, whether measured by R2, RSS, or the standard error of the regression (the estimate of the standard deviation of u, here denoted Root MSE), is likewise not affected by the change.





21

But the t tests are affected. In particular, the meaning of a null hypothesis for a dummy variable coefficient being equal to 0 is different.








22

For example, the t statistic for the technical school coefficient is for the null hypothesis that the overhead costs of technical schools are the same as those of skilled workers’ schools.


23

The t ratio in question is only 0.35, so the null hypothesis is not rejected.








24

The t ratio for the coefficient of VOC is –2.65, so one concludes that the overheads of vocational schools are significantly lower than those of skilled workers’ schools, at the 1% significance level.





25

General schools clearly have lower overhead costs than the skilled workers’ schools, according to the regression.


26

Note that there are some differences in the standard errors. The standard error of the coefficient of N is unaffected.

. reg COST N TECH WORKER VOC------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 342.6335 40.2195 8.519 0.000 262.3978 422.8692 TECH | 154110.9 26760.41 5.759 0.000 100725.3 207496.4 WORKER | 143362.4 27852.8 5.147 0.000 87797.57 198927.2 VOC | 53228.64 31061.65 1.714 0.091 -8737.646 115194.9 _cons | -54893.09 26673.08 -2.058 0.043 -108104.4 -1681.748------------------------------------------------------------------------------


. reg COST N TECH VOC GEN------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 342.6335 40.2195 8.519 0.000 262.3978 422.8692 TECH | 10748.51 30524.87 0.352 0.726 -50146.93 71643.95 VOC | -90133.74 33984.22 -2.652 0.010 -157930.4 -22337.07 GEN | -143362.4 27852.8 -5.147 0.000 -198927.2 -87797.57 _cons | 88469.29 28849.56 3.067 0.003 30916.01 146022.6------------------------------------------------------------------------------



27

The one test involving the dummy variables that can be performed with either specification is the test of whether the overhead costs of general schools and skilled workers’ schools are different.


28

The choice of specification can make no difference to the outcome of this test. The only difference is caused by the fact that the regression coefficient has become negative in the second specification.




29

The standard error is the same, so the t statistic has the same absolute magnitude and the outcome of the test must be the same.






30

However the standard errors of the coefficients of the other dummy variables are slightly larger in the second specification.


31

This is because the skilled workers’ schools are less ‘normal’ or ‘basic’ than the general schools and there are fewer of them in the sample (only 17, as opposed to 28).




32

As a consequence there is less precision in measuring the difference between their costs and those of the other schools than there was when general schools were the reference category.




Copyright Christopher Dougherty 2012.

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

http://www.oup.com/uk/orc/bin/9780199567089/.

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

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

EC2020 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2012.11.04

Documents

THE EFFECTS OF CHANGING THE REFERENCE CATEGORY 1 In the previous sequence we chose general academic schools as the reference (omitted) category and defined