COST 11 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES 1 This sequence explains how you can...

COST

1

DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

1

This sequence explains how you can include qualitative explanatory variables in your regression model.

N

regular schooloccupational school

1'

COST

1

2

Suppose 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.


N


1'

COST

1

3

The 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.


N


1'

4

One way of dealing with the difference in the costs would be to run separate regressions for the two types of school.


COST

1

N


1'

5

However 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.


COST

1

N


1'

6

Another way of handling the difference would be to hypothesize that the cost function for occupational schools has an intercept 1' that is greater than that for regular schools.


N

COST

1

OCC = 0 Regular school COST = 1 + 2N + u

OCC = 1 Occupational school COST = 1' + 2N + u

1'


7

Effectively, 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.




N

COST

1


1'

8

Let us define to be the difference in the intercepts: = 1' – 1.




Define = 1' – 1

N

COST

1


1'

9

Then 1' = 1 + and we can rewrite the cost function for occupational schools as shown.



OCC = 1 Occupational school COST = 1 + + 2N + u

Define = 1' – 1

1+

N

COST

1occupational schoolregular school



Combined equation COST = 1 + OCC + 2N + u

10

We 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.


N

COST

1

1+

occupational schoolregular school

Dummy 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.

11




Combined equation COST = 1 + OCC + 2N + u

N

COST

1

1+


We will now fit a function of this type using actual data for a sample of 74 secondary schools in Shanghai.

12


-100000

0

100000

200000

300000

400000

500000

600000

0 200 400 600 800 1000 1200

COST

N


School Type COST N OCC

1 Occupational 345,000 623 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




The 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.

13


14

OCC is the dummy variable for the type of school.


School Type COST N OCC



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




. reg COST N OCC

Source | SS df MS Number of obs = 74---------+------------------------------ F( 2, 71) = 56.86 Model | 9.0582e+11 2 4.5291e+11 Prob > F = 0.0000Residual | 5.6553e+11 71 7.9652e+09 R-squared = 0.6156---------+------------------------------ Adj R-squared = 0.6048 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 89248

------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- N | 331.4493 39.75844 8.337 0.000 252.1732 410.7254 OCC | 133259.1 20827.59 6.398 0.000 91730.06 174788.1 _cons | -33612.55 23573.47 -1.426 0.158 -80616.71 13391.61------------------------------------------------------------------------------

We now run the regression of COST on N and OCC, treating OCC just like any other explanatory variable, despite its artificial nature. The Stata output is shown.

15


We will begin by interpreting the regression coefficients.

16


. reg COST N OCC



17

The 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.


COST = –34,000 + 133,000OCC + 331N^

Regular school(OCC = 0)

COST = –34,000 + 133,000OCC + 331N

COST = –34,000 + 331N^

^

18

If 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.


19

Obviously 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.



COST = –34,000 + 133,000OCC + 331N

COST = –34,000 + 331N^

^

20

The coefficient of the dummy variable is an estimate of , the extra annual overhead cost of an occupational school.



COST = –34,000 + 133,000OCC + 331N

COST = –34,000 + 331N^

^


Putting 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.

21

COST = –34,000 + 133,000OCC + 331N

COST = –34,000 + 331N

COST = –34,000 + 133,000 + 331N

= 99,000 + 331N

^

^

^


Occupational school(OCC = 1)

The scatter diagram shows the data and the two cost functions derived from the regression results.

22


COST

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0

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0 200 400 600 800 1000 1200 N


In addition to the estimates of the coefficients, the regression results will include standard errors and the usual diagnostic statistics.

23


. reg COST N OCC



We will perform a t test on the coefficient of the dummy variable. Our null hypothesis is H0: = 0 and our alternative hypothesis is H1: 0.

24


. reg COST N OCC



In words, 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.

25


. reg COST N OCC



. reg COST N OCC



We can perform t tests on the other coefficients in the usual way. The t statistic for the coefficient of N is 8.34, so we conclude that the marginal cost is (very) significantly different from 0.

26


. reg COST N OCC



In the case of the intercept, the t statistic is –1.43, so we do not reject the null hypothesis H0: 1 = 0.

27


Thus one explanation of the nonsensical negative overhead cost of regular schools might be that they do not actually have any overheads and our estimate is a random number.

28


. reg COST N OCC



. reg COST N OCC



A more realistic version of this hypothesis is that 1 is positive but small (as you can see, the 95 percent confidence interval includes positive values) and the error term is responsible for the negative estimate.

29


As already noted, a further possibility is that the model is misspecified in some way. We will continue to develop the model in the next sequence.

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. reg COST N OCC



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 5.1 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

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COST 11 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES 1 This sequence explains how you can...

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