# Using an overall category as a reference group for dummy variables

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?

• I believe you are referring to effect coding. Dec 23, 2015 at 2:29
• Thanks for the reply, this looks useful. However, effect coding uses the grand mean of the groups as the intercept, so our coefficients tell us the effect of being in that group compared to the mean. But grand mean of the unemployment rate within education groups is not the same as the overall unemployment rate, which is what I would like to compare to. Dec 23, 2015 at 7:30

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.