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SLOPE DUMMY VARIABLES 1 The scatter diagram shows the data for the 74 schools in Shanghai and the cost functions derived from a regression of COST on N and a dummy variable for the type of curriculum (occupational / regular).

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SLOPE DUMMY VARIABLES 2 The specification of the model incorporates the assumption that the marginal cost per student is the same for occupational and regular schools. Hence the cost functions are parallel.

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SLOPE DUMMY VARIABLES 3 However, this is not a realistic assumption. Occupational schools incur expenditure on training materials that is related to the number of students.

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SLOPE DUMMY VARIABLES 4 Also, the staff-student ratio has to be higher in occupational schools because workshop groups cannot be, or at least should not be, as large as academic classes.

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SLOPE DUMMY VARIABLES 5 Looking at the scatter diagram, you can see that the cost function for the occupational schools should be steeper, and that for the regular schools should be flatter.

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SLOPE DUMMY VARIABLES 6 We will relax the assumption of the same marginal cost by introducing what is known as a slope dummy variable. This is NOCC, defined as the product of N and OCC. COST = 1 + OCC + 2 N + NOCC + u

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SLOPE DUMMY VARIABLES 7 In the case of a regular school, OCC is 0 and hence so also is NOCC. The model reduces to its basic components. COST = 1 + OCC + 2 N + NOCC + u Regular schoolCOST = 1 + 2 N + u (OCC = NOCC = 0)

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SLOPE DUMMY VARIABLES In the case of an occupational school, OCC is equal to 1 and NOCC is equal to N. The equation simplifies as shown. 8 COST = 1 + OCC + 2 N + NOCC + u Regular schoolCOST = 1 + 2 N + u (OCC = NOCC = 0) Occupational schoolCOST = ( 1 + ) + ( 2 + N + u (OCC = 1; NOCC = N)

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SLOPE DUMMY VARIABLES The model now allows the marginal cost per student to be an amount greater than that in regular schools, as well as allowing the overhead costs to be different. 9 COST = 1 + OCC + 2 N + NOCC + u Regular schoolCOST = 1 + 2 N + u (OCC = NOCC = 0) Occupational schoolCOST = ( 1 + ) + ( 2 + N + u (OCC = 1; NOCC = N)

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COST N 1 + 11 Occupational Regular SLOPE DUMMY VARIABLES The diagram illustrates the model graphically. 10

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SLOPE DUMMY VARIABLES Here are the data for the first ten schools. Note the weird way in which NOCC is defined. 11 School TypeCOST N OCC NOCC 1Occupational345,0006231623 2Occupational 537,0006531653 3Regular 170,00040000 4Occupational 526.0006631663 5Regular100,00056300 6Regular 28,00023600 7Regular 160,00030700 8Occupational 45,0001731173 9Occupational 120,0001461146 10 Occupational61,00099199

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. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- N | 152.2982 60.01932 2.537 0.013 32.59349 272.003 OCC | -3501.177 41085.46 -0.085 0.932 -85443.55 78441.19 NOCC | 284.4786 75.63211 3.761 0.000 133.6351 435.3221 _cons | 51475.25 31314.84 1.644 0.105 -10980.24 113930.7 ------------------------------------------------------------------------------ SLOPE DUMMY VARIABLES Weird or not, the procedure works very well. Here is the regression output using the full sample of 74 schools. We will begin by interpreting the regression coefficients. 12

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SLOPE DUMMY VARIABLES Here is the regression in equation form. 13 COST = 51,000 – 4,000 OCC + 152N + 284NOCC ^

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SLOPE DUMMY VARIABLES Putting OCC, and hence NOCC, equal to 0, we get the cost function for regular schools. We estimate that their annual overhead costs are 51,000 yuan and their annual marginal cost per student is 152 yuan. 14 COST = 51,000 – 4,000 OCC + 152N + 284NOCC Regular schoolCOST= 51,000 + 152N (OCC = NOCC = 0) ^ ^

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SLOPE DUMMY VARIABLES Putting OCC equal to 1, and hence NOCC equal to N, we estimate that the annual overhead costs of the occupational schools are 47,000 yuan and the annual marginal cost per student is 436 yuan. 15 COST = 51,000 – 4,000 OCC + 152N + 284NOCC Regular schoolCOST= 51,000 + 152N (OCC = NOCC = 0) Occupational schoolCOST= 51,000 – 4,000 + 152N + 284N (OCC = 1; NOCC = N) = 47,000 + 436N ^ ^ ^

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SLOPE DUMMY VARIABLES You can see that the cost functions fit the data much better than before and that the real difference is in the marginal cost, not the overhead cost. 16

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SLOPE DUMMY VARIABLES Now we can see why we had a nonsensical negative estimate of the overhead cost of a regular school in previous specifications. 17

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SLOPE DUMMY VARIABLES The assumption of the same marginal cost led to an estimate of the marginal cost that was a compromise between the marginal costs of occupational and regular schools. 18

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SLOPE DUMMY VARIABLES The cost function for regular schools was too steep and as a consequence the intercept was underestimated, actually becoming negative and indicating that something must be wrong with the specification of the model. 19

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SLOPE DUMMY VARIABLES We can perform t tests as usual. The t statistic for the coefficient of NOCC is 3.76, so the marginal cost per student in an occupational school is significantly higher than that in a regular school. 20. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- N | 152.2982 60.01932 2.537 0.013 32.59349 272.003 OCC | -3501.177 41085.46 -0.085 0.932 -85443.55 78441.19 NOCC | 284.4786 75.63211 3.761 0.000 133.6351 435.3221 _cons | 51475.25 31314.84 1.644 0.105 -10980.24 113930.7 ------------------------------------------------------------------------------

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SLOPE DUMMY VARIABLES The coefficient of OCC is now negative, suggesting that the overhead costs of occupational schools are actually lower than those of regular schools. 21. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- N | 152.2982 60.01932 2.537 0.013 32.59349 272.003 OCC | -3501.177 41085.46 -0.085 0.932 -85443.55 78441.19 NOCC | 284.4786 75.63211 3.761 0.000 133.6351 435.3221 _cons | 51475.25 31314.84 1.644 0.105 -10980.24 113930.7 ------------------------------------------------------------------------------

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SLOPE DUMMY VARIABLES This is unlikely. However, the t statistic is only -0.09, so we do not reject the null hypothesis that the overhead costs of the two types of school are the same. 22. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------ COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- N | 152.2982 60.01932 2.537 0.013 32.59349 272.003 OCC | -3501.177 41085.46 -0.085 0.932 -85443.55 78441.19 NOCC | 284.4786 75.63211 3.761 0.000 133.6351 435.3221 _cons | 51475.25 31314.84 1.644 0.105 -10980.24 113930.7 ------------------------------------------------------------------------------

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SLOPE DUMMY VARIABLES We can also perform an F test of the joint explanatory power of the dummy variables, comparing RSS when the dummy variables are included with RSS when they are not. 23. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The null hypothesis is that the coefficients of OCC and NOCC are both equal to 0. The alternative hypothesis is that one or both are nonzero. 24

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The improvement in the fit on adding the dummy variables is the reduction in RSS. 25

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The cost is 2 because 2 extra parameters, the coefficients of the dummy variables, have been estimated, and as a consequence the number of degrees of freedom remaining has been reduced from 72 to 70. 26

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The first component of the denominator is RSS after the dummies have been added. 27

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The denominator is RSS after the dummies have been added, divided by the number of degrees of freedom remaining. This is 70 because there are 74 observations and 4 parameters have been estimated. 28

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 The F statistic is therefore 31.4. The critical vale of F(2,70) at the 0.1 percent level is 7.6. 29

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SLOPE DUMMY VARIABLES. reg COST N OCC NOCC Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 3, 70) = 49.64 Model | 1.0009e+12 3 3.3363e+11 Prob > F = 0.0000 Residual | 4.7045e+11 70 6.7207e+09 R-squared = 0.6803 ---------+------------------------------ Adj R-squared = 0.6666 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 81980 ------------------------------------------------------------------------------. reg COST N Source | SS df MS Number of obs = 74 ---------+------------------------------ F( 1, 72) = 46.82 Model | 5.7974e+11 1 5.7974e+11 Prob > F = 0.0000 Residual | 8.9160e+11 72 1.2383e+10 R-squared = 0.3940 ---------+------------------------------ Adj R-squared = 0.3856 Total | 1.4713e+12 73 2.0155e+10 Root MSE = 1.1e+05 Thus we conclude that at least one of the dummy variable coefficients is different from 0. We knew this already from the t tests, so in this case the F test does not actually add anything. 30

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Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 24.06.06

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