Usually linear regression involves two variables (x,y), i.e. an independent variable x and a dependent...

Carole Juarez

Carole Juarez

Answered question

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=a0+a1x
where a0 and a1 are parameters that define the linear model. In linear regression we have one equation of this form for each couple of observed variables (xi,yi), thus we have a linear system and its solution gives us a0 and a1.
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=b0+b1w
where b0 and b1 are parameters that define the linear relation between z and w. Also in this case a set of observation (wj,zj) leads to a linear system and its solution gives us b0 and b1.
In general, if a0,a1 and b1 are independent, then we can solve the two linear systems separately. But now, let's suppose that a0 and b0 are independent, while a1=b1. 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 R2).

Answer & Explanation

Gene Espinosa

Gene Espinosa

Beginner2022-02-23Added 7 answers

If your model is
y=a0+c1x
z=b0+c1w
you can minimize
(a0+c1xy)2+(b0+c1wz)2
giving the equations
a0+c1xy=0,
b0+c1wz=0,
x(a0+c1xy)+w(b0+c1wz)=0.
Now solve this 3×3 system for a0,b0,c1.
The fit quality is still given by the ratio of the explained variance over the total variance.

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