# Regression based performance attribution with dummy variables

I am following some work to do with a regression based performance attribution.

The regression is a cross sectional one. The $$y$$ vector is the risk free return for say 1,000 companies. The $$X$$ matrix is made up of a constant, some factors such as book to price, momentum etc, say we 6 such factors (7 including the constant). Then for a stock's country there are dummy variables (twenty countries). So our matrix is 1000 x 27 (including the constant).

However I thought when you have dummy variables you would not use all of them, you would use $$n-1$$ because it introduces multicollinearity. Is the regression above mis-specified?

You are right that if you use binary dummy variables for $$n$$ possible values of some feature (the country in your case) you need only $$n-1$$ variables because the last (or first) country is indicated by all dummy variables equal to zero.