Dummy Variable

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Can someone explain why I can't code dummy variables this way?

I understand that If you have 4 groups, you need only 3 dummy variables, Otherwize you will have multicolinearity. You would normally code the dummies like this:

Eg:
3 groups A,B,C
D(1) = 1 for A and 0 for B and C
D(2) = 1 for B and 0 for A and C
D(3) = 1 for C and 0 for A and B

But what if I could code it this way?

D(1) = 0 AND D(2) = 0 for A

D(1) = 0 AND D(2) = 1 for B

D(1) = 1 AND D(2) = 0 for C

This way I use only 2 variables to represent 3 groups, so I gain degrees of freedom right??

But I haven't come across any good explanation for why I should not code the variables this way.

This is fine - if you don't really want to include A.

The problem with this is that you can't distinguish between A and the complement of (B or C). You could tweak the numbers a little to get around this, but fundamentally if you only have m dummy variables for n>m groups, you can only recover some set of m linear combinations of the n groups; you can't resolve this uniquely into n separate effects. So if you have n groups of interest, you really do need n dummy variables.

Chapter 14: Mult. Reg: Dummy variables and Interaction


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