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What kind of regression shall I use for such a data set where all explanatory variables are categorical (not ordinal) and dependent variable being ordinal? Can I use classification? Thanks in advance!

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Whether you should use a regression method or a classification method is for you to decide, based on what you are trying to do. There is some overlap among methods.

A good starting point for regression when the DV is ordinal is ordinal logistic regression. It does make some assumptions and if they are violated there are alternatives such as partial proportional odds or multinomial logistic regression. The latter gets unwieldy if there are a lot of levels to the DV.

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