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I have a dataset with several categorical variables. I have been running some regressions and used dummy coding for these categorical variables. The problem is that some specifications lead to perfect collinearity between a few vars when including all (but one) levels of a categorical variable of interest. Thus I need to omit two levels instead of one.

For instance, suppose variable $x_1$ is categorical and has four levels. Including the first three levels as dummies leads to perfect collinearity between the third level and variable $x_2$. Thus, the third level of $x_1$ is dropped for modeling purposes. Then variable $x_1$ has two instead of one reference levels.

This leads to statistically significant results, but I am not sure it is correct to do this at all. If so, is there any proper way to interpret the results?

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  • $\begingroup$ Welcome. You may not have a problem at all. What do these variables represent? Do you have any output to share? $\endgroup$ May 24 '21 at 23:50
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If a variable or one of its levels is perfectly contained in another one you are right to exclude it from the model. Proceed! The only problem you'll have is that you won't be able to tell if the variation in the dependent variable comes from variable x1 or x2 when they assume the collinear level. Try reading about instrumental variables or understand if there is a causal relationship between x1 and x2.

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