Regression reparametrisation Dummyvariables i have a question where I don't get the solution. Maybee it is not asked ans answered in internet as this is a common question to drop out students:
I have a model (linear regression) with a dummy variable that represents 6 categories. I know due to dummy variable trap I can use only 5 of them, the 6th (D_6) is in the intercept as reference kategory. The model does not have other effects.
Y_hat = ß0 + D_1*c_1 + D_2*c_2 + D_3*c_3 + D_4*c_4 + D_5*c_5

How do the coefficients of the model if D_1 is the new reference category?
My idea that the new model is
Y_hat = ß0 + D_2*c_2 + D_3*c_3 + D_4*c_4 + D_5*c_5 +D_6+c_6

Would make sense, but how are the change between the coefficients c?
Some Books says that all not affected c are still same, they don't change and the one that changes only changes the sign. Is that Correct?
 A: No, that is not correct: When you change the reference value for a categorical variable, the intercept term now becomes the 'effect size' for that value.  So in your second model the intercept parameter has now changed its meaning, and this flows through to change the meaning of all the slope coefficients.  In your first model you had:
$$\begin{array}{l} 
\beta_0 = \mathbb{E}(Y|X = 6), \\
c_1 = \mathbb{E}(Y|X = 1) - \mathbb{E}(Y|X = 6), \\
c_2 = \mathbb{E}(Y|X = 2) - \mathbb{E}(Y|X = 6), \\
c_3 = \mathbb{E}(Y|X = 3) - \mathbb{E}(Y|X = 6), \\
c_4 = \mathbb{E}(Y|X = 4) - \mathbb{E}(Y|X = 6), \\
c_5 = \mathbb{E}(Y|X = 5) - \mathbb{E}(Y|X = 6). \end{array}$$
In your second model you have:
$$\begin{array}{l} 
\beta_0 = \mathbb{E}(Y|X = 1), \\
c_2 = \mathbb{E}(Y|X = 2) - \mathbb{E}(Y|X = 1), \\
c_3 = \mathbb{E}(Y|X = 3) - \mathbb{E}(Y|X = 1), \\
c_4 = \mathbb{E}(Y|X = 4) - \mathbb{E}(Y|X = 1), \\
c_5 = \mathbb{E}(Y|X = 5) - \mathbb{E}(Y|X = 1), \\
c_6 = \mathbb{E}(Y|X = 6) - \mathbb{E}(Y|X = 1). \end{array}$$
