THE DUMMY VARIABLE TRAP

1

Suppose that you have a regression model with Y depending on a set of ordinary variables X2, ..., Xk and a qualitative variable.

uDDXXY sskk ...... 22221

2

Suppose that the qualitative variable has s categories. We choose one of them as the omitted category (without loss of generality, category 1) and define dummy variables D2, ..., Ds for the rest.

THE DUMMY VARIABLE TRAP

uDDXXY sskk ...... 22221

3

What would happen if we did not drop the reference category? Suppose we defined a dummy variable D1 for it and included it in the specification. What would happen then?

THE DUMMY VARIABLE TRAP

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

4

We would fall into the dummy variable trap. It would be impossible to fit the model as specified.

THE DUMMY VARIABLE TRAP

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

5

We will start with an intuitive explanation. The coefficient of each dummy variable represents the increase in the intercept relative to that for the basic category. But there is no basic category for such a comparison.

THE DUMMY VARIABLE TRAP

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

6

1 represents the fixed component of Y for the basic category. But again, there is no basic category. Thus the model does not have any logical interpretation.

THE DUMMY VARIABLE TRAP

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

7

Mathematically, we have a special case of exact multicollinearity. If there is no omitted category, there is an exact linear relationship between X1 and the dummy variables. The table gives an example where there are 4 categories.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

8

X1 is the variable whose coefficient is 1. It is equal to 1 in all observations. Usually we do not write it explicitly because there is no need to do so.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

9

If there is an exact linear relationship among a set of the variables, it is impossible in principle to estimate the separate coefficients of those variables. To understand this properly, one needs to use linear algebra.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

10

If you tried to run the regression anyway, the regression application should detect the problem and do one of two things. It may simply refuse to run the regression.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

11

Alternatively, it may run it, dropping one of the variables in the linear relationship, effectively defining the omitted category by itself.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

12

There is another way of avoiding the dummy variable trap. That is to drop the intercept (and X1). There is no longer a problem because there is no longer an exact linear relationship linking the variables.

THE DUMMY VARIABLE TRAP

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

uDDDXXY sskk ...... 221122

13

The parameters are now the intercepts in the relationship for the individual categories. For example, if the observation relates to category 2, all the dummy variables except D2 will be equal to 0. D2 = 1, and hence the relationship for that observation has intercept 2.

Observation Category X1 D1 D2 D3 D4

1 4 1 0 0 0 12 3 1 0 0 1 03 1 1 1 0 0 04 2 1 0 1 0 05 2 1 0 1 0 06 3 1 0 0 1 07 1 1 1 0 0 08 4 1 0 0 0 1

THE DUMMY VARIABLE TRAP

1

4

1

XDi

i

uDDXXY sskk ...... 22221

uDDDXXY sskk ...... 2211221

uDDDXXXY sskk ...... 22112211

uDDDXXY sskk ...... 221122

Copyright Christopher Dougherty 2012.

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2012.11.05