# Can you leave out more than one dummy variable to have more than one variables in the reference category?

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?

• Can you explain why do you want this? – kjetil b halvorsen Apr 8 '15 at 11:53
• 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. – ttnphns Apr 8 '15 at 11:56
• The motivation was to save degrees of freedom. Otherwise, I realise it would have been best to include all k-1 (9) dummy variables. – Tom Markovitch Apr 8 '15 at 23:16