Usually linear regression involves two variables (x,y), i.e. an independent variable x and a dependent variable y, and they are related by the following expression
where and are parameters that define the linear model. In linear regression we have one equation of this form for each couple of observed variables , thus we have a linear system and its solution gives us and .
Let's consider that we have two set of independent-dependent variables, namely (x,y) and (w,z). The first two variables (x,y) are related by the previous equation, while the second two variables (w,z) are related by the following
where and are parameters that define the linear relation between z and w. Also in this case a set of observation leads to a linear system and its solution gives us and .
In general, if and are independent, then we can solve the two linear systems separately. But now, let's suppose that and are independent, while . In this case, the two linear systems should be solved simultaneously.
I've solved this problem just definying one linear system of equation involving both the two sets of equations, but I would like to know if this problem has a specific name and how to correctly approach it. In particular, I want to know how to assessing the fit quality (for example, with an equivalent of the ).