When using a categorical variable in a multiple regression model that has k levels, how many dummy variables are needed?

In a multiple regression model, when incorporating a categorical variable with k levels, you only need k - 1 dummy variables to properly represent the categorical variable. This is because one level of the categorical variable can be used as a baseline or reference category, which can be left out of the model. This approach avoids the problem of multicollinearity, which can arise when you include all levels as separate dummy variables.

By using k - 1 dummy variables, each remaining variable effectively compares the respective level of the categorical variable to the baseline category. For example, if you have a categorical variable with three levels—let's say "Red," "Blue," and "Green"—you would create two dummy variables. One dummy could represent whether the observation is "Red" or not, and the other could represent whether it is "Blue" or not. The "Green" category would be implicitly represented when both dummy variables are zero.

This method allows for efficient modeling while maintaining interpretability. Utilizing all k dummy variables would lead to redundancy in the model, which is why k - 1 is the correct number needed.