Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: the dummy variable trap Original citation: Dougherty, C. (2012) EC220

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 5)Slideshow: the dummy variable trap

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 5). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/131/

Available in LSE Learning Resources Online: May 2012

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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.



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?


uDDDXXY sskk ...... 2211221


4

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




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.




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.




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.



uDDDXXXY sskk ...... 22112211

1

4

1

XDi

i

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


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.




1

4

1

XDi

i


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


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.




1

4

1

XDi

i


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


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.




1

4

1

XDi

i


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


11

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




1

4

1

XDi

i


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


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.




1

4

1

XDi

i



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


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.




1

4

1

XDi

i



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

Copyright Christopher Dougherty 2011.

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

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own and 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

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

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Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: the dummy variable trap Original citation: Dougherty, C. (2012) EC220