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For example, in the simple OLS regression:

  1. $y = a + b_1x_1 + ... + b_kx_k + \varepsilon$

if your dummy variable $d$ has 10 categories, could you include just one dummy variable for instance:

  1. $y = a + b_1x_1 + ... + b_kx_k + B_1d_1 + \varepsilon$

and even interact with just one dummy variable for instance:

  1. $y = a + b_1x_1 + ... + b_kx_k + B_1d_1 + B_2d_1x_1 + \varepsilon$

If this is possible, is the interpretation of $B_1$ the effect of being in the $d_1$ category relative to all other categories (and what does this quite mean - I think it is the average distance between the slopes of the other categories with the $d_1$ category?)

Likewise, would this make the interpretation of $B_2$ the additional effect of $x_1$ from being in the $d_1$ category relative to all other categories?

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  • $\begingroup$ Can you explain why do you want this? $\endgroup$
    – kjetil b halvorsen ♦
    Commented Apr 8, 2015 at 11:53
  • $\begingroup$ Imagine that when you use only one of the 10 dummies in the model it is the same as if your initial categorical variable were 2-category rather than 10-category. $\endgroup$
    – ttnphns
    Commented Apr 8, 2015 at 11:56
  • $\begingroup$ The motivation was to save degrees of freedom. Otherwise, I realise it would have been best to include all k-1 (9) dummy variables. $\endgroup$
    – Tom Markovitch
    Commented Apr 8, 2015 at 23:16

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