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As we know, the AIC is defined as $\mathrm{AIC} = 2k - 2\ln(L)$ with $k$ being the amount of estimated parameters.

My question is: in case I have a categorical predictor, lets say: educational backgroud, with three categories. Does this count as 1 or as 3 estimated parameters in the AIC's sense?

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In this case you require two dummy variables to be coded to represent two of the three categories, with the last category's effect absorbed into the intercept of the regression.

That is, suppose educational background takes values low, medium, high, then you might end up with two binary dummy variables: isLow, isMedium. Thus, the number of parameters k is 2.

More information can be found in this Q/A: Do dummy variables count as independent variables when calculating degrees of freedom in a multiple regression?

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