1. COST 11 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES 1 This sequence explains how you can include qualitative explanatory variables in your regression model. N regular school occupational school 1'1'

2. COST 1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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES N regular school occupational school 1'1'

3. COST 1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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES N regular school occupational school 1'1'

4. 4 One way of dealing with the difference in the costs would be to run separate regressions for the two types of school. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES COST 11 N regular school occupational school 1'1'

5. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES COST 11 N regular school occupational school 1'1'

6. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES N COST 11 OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 ' +  2 N + u 1'1' regular school occupational school

7. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 ' +  2 N + u N COST 11 regular school occupational school 1'1'

8. 8 Let us define  to be the difference in the intercepts:  =  1 ' –  1. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 ' +  2 N + u Define  =  1 ' –  1 N COST 11 regular school occupational school  1'1'

9. 9 Then  1 ' =  1 +  and we can rewrite the cost function for occupational schools as shown. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 +  +  2 N + u Define  =  1 ' –  1 1+1+ N COST 11 occupational school regular school 

10. OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 +  +  2 N + u Combined equationCOST =  1 +  OCC +  2 N + 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES N COST 11 1+1+ occupational school regular school 

11. 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES OCC = 0 Regular schoolCOST =  1 +  2 N + u OCC = 1 Occupational schoolCOST =  1 +  +  2 N + u Combined equationCOST =  1 +  OCC +  2 N + u N COST 11 1+1+ occupational school regular school 

12. We will now fit a function of this type using actual data for a sample of 74 secondary schools in Shanghai. 12 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES COST N occupational school regular school

13. School Type COST N OCC 1Occupational345,0006231 2Occupational 537,0006531 3Regular 170,0004000 4Occupational 526.0006631 5Regular100,0005630 6Regular 28,0002360 7Regular 160,0003070 8Occupational 45,0001731 9Occupational 120,0001461 10 Occupational61,000991 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

14. 14 OCC is the dummy variable for the type of school. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES School Type COST N OCC 1Occupational345,0006231 2Occupational 537,0006531 3Regular 170,0004000 4Occupational 526.0006631 5Regular100,0005630 6Regular 28,0002360 7Regular 160,0003070 8Occupational 45,0001731 9Occupational 120,0001461 10 Occupational61,000991

15. . 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 = 0.0000 Residual | 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

16. We will begin by interpreting the regression coefficients. 16 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

17. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES COST = –34,000 + 133,000OCC + 331N ^

18. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

19. 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. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES Regular school (OCC = 0) COST = –34,000 + 133,000OCC + 331N COST = –34,000 + 331N ^ ^

20. 20 The coefficient of the dummy variable is an estimate of , the extra annual overhead cost of an occupational school. DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES Regular school (OCC = 0) COST = –34,000 + 133,000OCC + 331N COST = –34,000 + 331N ^ ^

21. Regular school (OCC = 0) 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 ^ ^ ^ DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES Occupational school (OCC = 1)

22. The scatter diagram shows the data and the two cost functions derived from the regression results. 22 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES COST N occupational school regular school

23. In addition to the estimates of the coefficients, the regression results will include standard errors and the usual diagnostic statistics. 23 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

24. We will perform a t test on the coefficient of the dummy variable. Our null hypothesis is H 0 :  = 0 and our alternative hypothesis is H 1 :  0. 24 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

25. 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

26. . 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 = 0.0000 Residual | 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 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

27. . 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------ In the case of the intercept, the t statistic is –1.43, so we do not reject the null hypothesis H 0 :  1 = 0. 27 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

28. 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

29. . 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------ 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 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES

30. 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. 30 DUMMY VARIABLE CLASSIFICATION WITH TWO CATEGORIES. 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 = 0.0000 Residual | 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 ------------------------------------------------------------------------------

31. 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/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 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/lsewww.londoninternational.ac.uk/lse. 2012.11.04