If we have X and Y that are mathematically dependent: X = Y + Z, is it 'forbidden' to use Y as a predictor to X in linear regression? I'm trying to find a concise explanation for why it is, or isn't.
One explanation I've found is that you shouldn't use simple linear regression in this case, but multiple linear regression, with Y and Z as predictors, but this doesn't seem right. The mathematical relationship is exact, and the multiple regression would return coefficients 0, 1, and 1 for X = 0 + 1Y + 1Z. So, there is something more fundamental I'm missing.
The purpose for linear regression is predicting the value of one variable from the other. But we already know that X can be calculated as the sum of Y and Z, so effectively, we are calculating the regression between Y and Y + Z. Is this 'forbidden'?
But, if we can only measure Y, does the regression give us a tool to predict X? Isn't it better to find the regression between Y and Z? I'm confused.