I think that when we move from the first model to the second, we need to adjust the first "intercept" term to be the average of all rows except Doctor rows, to really have a better fitting (least squares error) model than before: income = 15 + 35 * Dr_i. Including the cleaner variable gives us: income = 20 + 30 * Dr_i - 7.5 * Cleaner_i. Notice that the first term is 20, which is already the lawyer income and adding the lawyer variable is superfluous and would make the model impossible to fit.
You're right, the intercept does change. I build a model in R and came up with the same equation as you for the second model. For the third model you don't need a lawyer variable because you always use one less dummy variable than you have categories.
I think that when we move from the first model to the second, we need to adjust the first "intercept" term to be the average of all rows except Doctor rows, to really have a better fitting (least squares error) model than before: income = 15 + 35 * Dr_i. Including the cleaner variable gives us: income = 20 + 30 * Dr_i - 7.5 * Cleaner_i. Notice that the first term is 20, which is already the lawyer income and adding the lawyer variable is superfluous and would make the model impossible to fit.
I think you've missed the point there.
@@jimbocho660 That is possible, but your comment isn't particularly helpful I'm afraid..
You're right, the intercept does change. I build a model in R and came up with the same equation as you for the second model. For the third model you don't need a lawyer variable because you always use one less dummy variable than you have categories.