Yes, and then you can figure out different values of a and b for a different teacher, different class, different year, different town, etc.
And you're really left with c*teacher_skill + noise, where the noise in the teacher_skill measure may be what results in some good teachers getting bad ratings.
In my view, empirical modeling assumes that there is some underlying regularity that can be modeled, but this assumption seems questionable. And part of the process has to be to decide an acceptable level of good teachers getting bad ratings.
And you're really left with c*teacher_skill + noise, where the noise in the teacher_skill measure may be what results in some good teachers getting bad ratings.
In my view, empirical modeling assumes that there is some underlying regularity that can be modeled, but this assumption seems questionable. And part of the process has to be to decide an acceptable level of good teachers getting bad ratings.