Carole Juarez

2022-02-22

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
$y={a}_{0}+{a}_{1}x$
where ${a}_{0}$ and ${a}_{1}$ are parameters that define the linear model. In linear regression we have one equation of this form for each couple of observed variables $\left({x}_{i},{y}_{i}\right)$, thus we have a linear system and its solution gives us ${a}_{0}$ and ${a}_{1}$.
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
$z={b}_{0}+{b}_{1}w$
where ${b}_{0}$ and ${b}_{1}$ are parameters that define the linear relation between z and w. Also in this case a set of observation $\left({w}_{j},{z}_{j}\right)$ leads to a linear system and its solution gives us ${b}_{0}$ and ${b}_{1}$.
In general, if ${a}_{0},{a}_{1}$ and ${b}_{1}$ are independent, then we can solve the two linear systems separately. But now, let's suppose that ${a}_{0}$ and ${b}_{0}$ are independent, while ${a}_{1}={b}_{1}$. 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 ${R}^{2}$).

Gene Espinosa

$y={a}_{0}+{c}_{1}x$
$z={b}_{0}+{c}_{1}w$
you can minimize
$\sum {\left({a}_{0}+{c}_{1}x-y\right)}^{2}+\sum {\left({b}_{0}+{c}_{1}w-z\right)}^{2}$
giving the equations
$\sum {a}_{0}+{c}_{1}x-y=0$,
$\sum {b}_{0}+{c}_{1}w-z=0$,
$\sum x\left({a}_{0}+{c}_{1}x-y\right)+\sum w\left({b}_{0}+{c}_{1}w-z\right)=0$.
Now solve this $3×3$ system for ${a}_{0},{b}_{0},{c}_{1}$.
The fit quality is still given by the ratio of the explained variance over the total variance.

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