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

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY N COST

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? THE EFFECTS OF CHANGING THE REFERENCE CATEGORY N COST

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY N COST

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY N COST

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY COST =  1  +  T TECH +  V VOC +  G GEN +  2 N + 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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST =  1  +  T TECH +  V VOC +  G GEN +  2 N + u COST =  1  +  2 N + u

8 In the case of observations relating to technical schools, TECH is equal to 1 and the intercept increases by an amount  T. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST =  1  +  T TECH +  V VOC +  G GEN +  2 N + u COST =  1  +  2 N + u COST = (  1  +  T ) +  2 N + u Technical 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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST =  1  +  T TECH +  V VOC +  G GEN +  2 N + u COST =  1  +  2 N + u COST = (  1  +  T ) +  2 N + u Technical school (TECH = 1; VOC = GEN = 0)

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST =  1  +  T TECH +  V VOC +  G GEN +  2 N + u COST =  1  +  2 N + u COST = (  1  +  T ) +  2 N + u Technical school (TECH = 1; VOC = GEN = 0) COST = (  1  +  G ) +  2 N + u General school (GEN = 1; TECH = VOC = 0) COST = (  1  +  V ) +  2 N + u Vocational 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+T 1+V1+V Technical Workers’ Vocational General GG VV TT 11 1+G1+G THE EFFECTS OF CHANGING THE REFERENCE CATEGORY COST

12 Here are the data for the first 10 of the 74 schools with skilled workers’ schools as the reference category. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY School TypeCOST N TECH VOCGEN 1Technical345, Technical 537, General 170, Workers’ General 100, Vocational 28, Vocational 160, Technical 45, Technical 120, Workers’ 61,

. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | Here is the Stata output for the regression. We will focus first on the regression coefficients. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

14 The regression result is shown written as an equation. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY COST = 88, ,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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST = 88, ,000TECH – 90,000VOC – 143,000GEN + 343N ^ ^ COST= 88, N

16 Putting TECH equal to 1 and VOC and GEN equal to 0, we obtain the equation for the technical schools. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST = 88, ,000TECH – 90,000VOC – 143,000GEN + 343N Technical school (TECH = 1; VOC = GEN = 0) ^ ^ COST= 88, N ^ COST= 88, , N = 99, N

17 And similarly we obtain the equations for the vocational and general schools, putting VOC and GEN equal to 1 in turn. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST = 88, ,000TECH – 90,000VOC – 143,000GEN + 343N Technical school (TECH = 1; VOC = GEN = 0) General school (VOC = 1; TECH = WORKER = 0) Vocational school (VOC = 1; TECH = GEN = 0) ^ ^ COST= 88, N ^ COST= 88, , N = 99, N ^ COST= 88,000 – 90, N = –2, N ^ COST= 88,000 – 143, N = –55, N

18 Note that the cost functions turn out to be exactly the same as when we used general schools as the reference category. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY Skilled workers' school (TECH = VOC = GEN = 0) COST = 88, ,000TECH – 90,000VOC – 143,000GEN + 343N Technical school (TECH = 1; VOC = GEN = 0) General school (VOC = 1; TECH = WORKER = 0) Vocational school (VOC = 1; TECH = GEN = 0) ^ ^ COST= 88, N ^ COST= 88, , N = 99, N ^ COST= 88,000 – 90, N = –2, N ^ COST= 88,000 – 143, N = –55, N

19 Consequently the scatter diagram with regression lines is exactly the same as before. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY N COST

20 The goodness of fit, whether measured by R 2, 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.. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | 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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

23 The t ratio in question is only 0.35, so the null hypothesis is not rejected. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | 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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

. reg COST N TECH VOC GEN Source | SS df MS Number of obs = F( 4, 69) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | General schools clearly have lower overhead costs than the skilled workers’ schools, according to the regression. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

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 | TECH | WORKER | VOC | _cons | THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | 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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons | However the standard errors of the coefficients of the other dummy variables are slightly larger in the second specification. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY

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). THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

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. THE EFFECTS OF CHANGING THE REFERENCE CATEGORY. reg COST N TECH WORKER VOC COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | WORKER | VOC | _cons | reg COST N TECH VOC GEN COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | TECH | VOC | GEN | _cons |

Copyright Christopher Dougherty 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 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 or the University of London International Programmes distance learning course EC2020 Elements of Econometrics