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If a model with a multi-level categorical variable is given in the following form:

$$logit(\pi_i) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3$$

Where $X_2$ and $X_3$ are the levels $2$ and $3$ from a categorical variable taking values $\in \{1,2,3\}$.

Then does this mean that value $1$ from that variable has been omitted or that it's in the model, but has been merged with the Intercept-term $\beta_0$?

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When you create a dummy variable with N levels, you only need (N-1) dummy variables to represent it.

For example, if you have (sunny, cloudy, rainy) as your levels, then knowing that sunny=0 and cloudy=0 implies rainy=1. Hence, it is not considered in the regression.

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