# EC220 - Introduction to econometrics (chapter 5)

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EC220 - Introduction to econometrics (chapter 5)
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: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

THE DUMMY VARIABLE TRAP
Suppose that you have a regression model with Y depending on a set of ordinary variables X2, ..., Xk and a qualitative variable. 1

THE DUMMY VARIABLE TRAP
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. 2

THE DUMMY VARIABLE TRAP
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? 3

THE DUMMY VARIABLE TRAP
We would fall into the dummy variable trap. I would be impossible to fit the model as specified. 4

THE DUMMY VARIABLE TRAP
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. 5

THE DUMMY VARIABLE TRAP
b1 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. 6

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 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. 7

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 X1 is the variable whose coefficient is b1. It is equal to 1 in all observations. Usually we do not write it explicitly because there is no need to do so. 8

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 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. 9

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 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. 10

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 Alternatively, it may run it, dropping one of the variables in the linear relationship, effectively defining the omitted category by itself. 11

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 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. 12

THE DUMMY VARIABLE TRAP
Observation Category X1 D1 D2 D3 D4 The d 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 d2. 13