interpretation on the regression with categorical variables

I have a regression of the following:

ax + bx + cx + Ax + Bx + Cx + intercept = y

where all x are dummy variables. This represents that I have 2 categorical variables each having 4 categories. So, if a=0, b=0, c=0 -> it means the fourth category of this is 1 automatically. In the same manner, if A=0, B=0, C=0 -> the other type of fourth category of this is 1 automatically.

If I want to simply get the effect of the fourth category from a, b, c can I simply set a=b=c=0 and get the intercept value as its fourth category value? If so, I am afraid that the fourth category variable of the other set when I do A=B=C=0 will give the same value as the intercept which I do not think making any sense.

May I get help on the interpretation?

• The coefficients in this model represent estimated mean differences from the baseline (i.e. fourth) category. The estimated mean difference of the fourth category from itself is 0. – Glen_b -Reinstate Monica Jul 22 '17 at 3:48

Here, $y$ is equal to the intercept when you have the fourth category in both variables.
Hence, there isn't a problem. Note that one thing is we think of "intercepts" as "y-intercept for a line," but that is for numerical predictors. Since here you only have two categorical variables, there are only 16 possible values for $y$.
• The effect of the fourth variable, reiative to the other variables, is given by the $a$, $b$, and $c$ coefficients - if you'd like to predict $y$, you would need to know the mean value attained by the second categorical variable (or have some sort of idea what value to put in; perhaps it could be a mode instead). – Kevin Jul 22 '17 at 15:35