Casualty Actuarial Society MAS-1 Practice Exam

When coding a categorical predictor with w levels, you use w-1 binary indicators to keep the model estimable. If you included all w dummies plus an intercept, the columns would be perfectly linearly related: for every observation the dummies sum to one, so one column would be a linear combination of the others and the intercept. This creates perfect multicollinearity, making the coefficients non-identifiable. By leaving out one category (the reference), the intercept plus the w-1 dummies provide a full-rank design matrix. The intercept captures the baseline category, and each included dummy measures the difference from that baseline. This is why the w-1 coding is used. The other ideas—increasing complexity, improving sparsity, or changing the response scale—don’t address the identifiability issue.