DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES This sequence explains how to extend the dummy variable technique to handle a qualitative explanatory variable which has more than two categories. 1 COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

In the previous sequence we used a dummy variable to differentiate between regular and occupational schools when fitting a cost function. 2 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

In actual fact there are two types of regular secondary school in Shanghai. There are general schools, which provide the usual academic education, and vocational schools. 3 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

As their name implies, the vocational schools are meant to impart occupational skills as well as give an academic education. 4 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

However the vocational component of the curriculum is typically quite small and the schools are similar to the general schools. Often they are just general schools with a couple of workshops added. 5 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

Likewise there are two types of occupational school. There are technical schools training technicians and skilled workers’ schools training craftsmen. 6 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

So now the qualitative variable has four categories. The standard procedure is to choose one category as the reference category and to define dummy variables for each of the others. 7 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

In general it is good practice to select the most normal or basic category as the reference category, if one category is in some sense more normal or basic than the others. 8 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

In the Shanghai sample it is sensible to choose the general schools as the reference category. They are the most numerous and the other schools are variations of them. 9 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

Accordingly we will define dummy variables for the other three types. TECH will be the dummy for the technical schools: TECH is equal to 1 if the observation relates to a technical school, 0 otherwise. 10 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

Similarly we will define dummy variables WORKER and VOC for the skilled workers’ schools and the vocational schools. 11 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

Each of the dummy variables will have a coefficient which represents the extra overhead costs of the schools, relative to the reference category. 12 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

Note that you do not include a dummy variable for the reference category, and that is the reason that the reference category is usually described as the omitted category. 13 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u

If an observation relates to a general school, the dummy variables are all 0 and the regression model is reduced to its basic components. 14 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u COST =  1  +  2 N + u

If an observation relates to a technical school, TECH will be equal to 1 and the other dummy variables will be 0. The regression model simplifies as shown. 15 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u COST =  1  +  2 N + u COST = (  1  +  T ) +  2 N + u Technical school (TECH = 1; WORKER = VOC = 0)

The regression model simplifies in a similar manner in the case of observations relating to skilled workers’ schools and vocational schools. 16 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST =  1  +  T TECH +  W WORKER +  V VOC +  2 N + u COST =  1  +  2 N + u COST = (  1  +  T ) +  2 N + u Technical school (TECH = 1; WORKER = VOC = 0) COST = (  1  +  V ) +  2 N + u Vocational school (VOC = 1; TECH = WORKER = 0) COST = (  1  +  W ) +  2 N + u Skilled workers' school (WORKER = 1; TECH = VOC = 0)

The diagram illustrates the model graphically. The  coefficients are the extra overhead costs of running technical, skilled workers’, and vocational schools, relative to the overhead cost of general schools. 17 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES N WW VV TT Workers’ Vocational Technical General 1+T1+T 1+W1+W 1+V1+V 11 COST

Note that we do not make any prior assumption about the size, or even the sign, of the  coefficients. They will be estimated from the sample data. 18 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES N WW VV TT Workers’ Vocational Technical General 1+T1+T 1+W1+W 1+V1+V 11 COST

School TypeCOST N TECH WORKERVOC 1Technical345, Technical 537, General 170, Workers’ General 100, Vocational 28, Vocational 160, Technical 45, Technical 120, Workers’ 61, Here are the data for the first 10 of the 74 schools. Note how the values of the dummy variables TECH, WORKER, and VOC are determined by the type of school in each observation. 19 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

The scatter diagram shows the data for the entire sample, differentiating by type of school. 20 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES N COST

Here is the Stata output for this regression. The coefficient of N indicates that the marginal cost per student per year is 343 yuan. 21 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons |

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | The coefficients of TECH, WORKER, and VOC are 154,000, 143,000, and 53,000, respectively, and should be interpreted as the additional annual overhead costs, relative to those of general schools. 22 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | The constant term is –55,000, indicating that the annual overhead cost of a general academic school is –55,000 yuan per year. Obviously this is nonsense and indicates that something is wrong with the model. 23 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

The top line shows the regression result in equation form. We will derive the implicit cost functions for each type of school. 24 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N ^

In the case of a general school, the dummy variables are all 0 and the equation reduces to the intercept and the term involving N. 25 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N ^ ^ COST= –55, N

The annual marginal cost per student is estimated at 343 yuan. The annual overhead cost per school is estimated at –55,000 yuan. Obviously a negative amount is inconceivable. 26 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N ^ ^ COST= –55, N

The extra annual overhead cost for a technical school, relative to a general school, is 154,000 yuan. Hence we derive the implicit cost function for technical schools. 27 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N Technical school (TECH = 1; WORKER = VOC = 0) ^ ^ COST= –55, N ^ COST= –55, , N = 99, N

And similarly the extra overhead costs of skilled workers’ and vocational schools, relative to those of general schools, are 143,000 and 53,000 yuan, respectively. 28 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N Technical school (TECH = 1; WORKER = VOC = 0) Vocational school (VOC = 1; TECH = WORKER = 0) Skilled workers' school (WORKER = 1; TECH = VOC = 0) ^ ^ COST= –55, N ^ COST= –55, , N = 99, N ^ COST= –55, , N = 88, N ^ COST= –55, , N = –2, N

Note that in each case the annual marginal cost per student is estimated at 343 yuan. The model specification assumes that this figure does not differ according to type of school. 29 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES General school (TECH = WORKER = VOC = 0) COST = –55, ,000TECH + 143,000WORKER + 53,000VOC + 343N Technical school (TECH = 1; WORKER = VOC = 0) Vocational school (VOC = 1; TECH = WORKER = 0) Skilled workers' school (WORKER = 1; TECH = VOC = 0) ^ ^ COST= –55, N ^ COST= –55, , N = 99, N ^ COST= –55, , N = 88, N ^ COST= –55, , N = –2, N

The four cost functions are illustrated graphically. 30 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES N COST

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | We can perform t tests on the coefficients in the usual way. The t statistic for N is 8.52, so the marginal cost is (very) significantly different from 0, as we would expect. 31 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | The t statistic for the technical school dummy is 5.76, indicating the the annual overhead cost of a technical school is (very) significantly greater than that of a general school, again as expected. 32 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | Similarly for skilled workers’ schools, the t statistic being DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons | In the case of vocational schools, however, the t statistic is only 1.71, indicating that the overhead cost of such a school is not significantly greater than that of a general school. 34 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

This is not surprising, given that the vocational schools are not much different from the general schools. 35 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons |

Note that the null hypotheses for the tests on the coefficients of the dummy variables are than the overhead costs of the other schools are not different from those of the general schools. 36 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons |

Finally we will perform an F test of the joint explanatory power of the dummy variables as a group. The null hypothesis is H 0 :  T =  W =  V = 0. The alternative hypothesis is that at least one  is different from DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons |

The residual sum of squares in the specification including the dummy variables is 5.41× DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N TECH WORKER VOC 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 | WORKER | VOC | _cons |

. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | _cons | The residual sum of squares in the specification excluding the dummy variables is 8.92× DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES

T he reduction in RSS when we include the dummies is therefore (8.92 – 5.41)× We will check whether this reduction is significant with the usual F test. 40 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

The numerator in the F ratio is the reduction in RSS divided by the cost, which is the 3 degrees of freedom given up when we estimate three additional coefficients (the coefficients of the dummies). 41 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

The denominator is RSS for the specification including the dummy variables, divided by the number of degrees of freedom remaining after they have been added. 42 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

The F ratio is therefore DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

F tables do not give the critical value for 3 and 69 degrees of freedom, but it must be lower than the critical value with 3 and 60 degrees of freedom. This is 6.17, at the 0.1% significance level. 44 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

Thus we reject H 0 at a high significance level. This is not exactly surprising since t tests show that TECH and WORKER have highly significant coefficients. 45 DUMMY CLASSIFICATION WITH MORE THAN TWO CATEGORIES. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e reg COST N TECH WORKER VOC 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 =

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