Using an overall category as a reference group for dummy variables

Question

I have data on the unemployment rate within 3 education groups for different states, and some other continuous data on for the given states e.g. GDP. I also have the overall unemployment rate for the state.

Each education dummy represents whether the observed unemployment rate is for the educational group 0-6 years of schooling, 7-12 years of schooling, and 13+ years of schooling.

I am regressing the unemployment rate on (a) educational dummies 1 2 and 3 and (b) my other continuous variables that don't vary by educational group e.g. GDP.

My regression has overall unemployment rate as the base group, so the coefficients of the dummy variables represent the difference between unemployment overall and unemployment within a given education group.

Is this a sensible way to choose the base group? Would issues of multicollinearity arise when looking at data on unemployment across the three groups and on the state as a whole? Would this issue disappear if there was more educational classifications and lots of variance in the data?

Answer 1

As long as you are using linear models, however you encode a categorical variable like educational group, the model is the same (in the sense that it will give identical predictions with identical input.) So while choosing an encoding that gives more interpretable coefficients is useful, you can just express the comparisons you want with contrasts.

Or, in your specific case, if you are using treatment coding for your categorical variable with three levels (as is default in R,) that is, if the group means are $\mu_1, \mu_2, \mu_3$, then group 1 is the reference group so the two parameters are $\beta_2 = \mu_2-\mu_1, \beta_3=\mu_3-\mu_1$. You want to compare with the general mean $\mu=\frac{n_1}{n}\mu_1 + \frac{n_2}{n}\mu_2 + \frac{n_3}{n}\mu_3$.

If you want the comparison, say, $\mu_3-\mu$, then some algebra will show that $$ \mu_3-\mu = \frac{n_1+n_2}{n}\beta_3-\frac{n_2}{n}\beta_2 $$ so the contrast vector to use is $(\dotsc,-\frac{n_2}{n},\frac{n_1+n_2}{n},\dotsc)$, the dots representing the other parameters in the model, which do not enter into the comparison, so should have zeros there.