Suppose we have three explanatory variables like $x_1,x_2,x_3$ and three response variables like $y_1, y_2, y_3$, we know that y should be a function of x, such that $$ (y_1,y_2,y_3) = f(x_1,x_2,x_3) $$ So I should make a regression model to find the pattern between x and y.
But the problem is that I want to predict $x_3$ instead of predicting y. After creating the above model, I want to do some optimization:
At the prediction step, given the best y, the observable variable $x_1$ and $x_2$, solving the best $x_3$ according to the regression model.
It seems that it's hard to solve it because y is multivariate. But on the other hand, we can treat $x_3$ as a function of $y, x_1, x_2$, such that $$ x_3 = f(y, x_1,x_2) $$ and make a regular regression model. My question is that is this method a correct way to do this? or do we have better method?