Linear models when responses have no link

I am studying normal linear regression and wanted to ask a question about its utility when working with independent RV. Suppose that we have for $$k \in [1,\dots,n]$$, $$Y_k = \beta_0 + \beta_1x_{k1} + \beta_2x_{k2} + \dots + \beta_qx_{kq} + \epsilon_k \quad \text{where } \epsilon_k \sim_{ind} N(0,\sigma^2)$$

We can write put all $$Y_k$$ in a vector $$Y$$, put the explanatory variables in a matrix $$X$$ and parameters in a vector $$\beta$$ and write $$Y=X\beta + \epsilon$$ where the disturbance vector $$\epsilon \sim N(0,\sigma^2 Id)$$.

However, in real life thinking, I do not see an utility to this matrix form if we are studying things that have no link. If $$Y_1$$ is the points you receive when asking a question in this website, $$Y_2$$ the salary of employees in a company, $$Y_3$$ the numbers of goals scored by a striker then these three only share parameters $$\beta_i$$ and explanatory variables can be totally different. So why do we choose to study them jointly ?

• $Y_1$, $Y_2$, $Y_3$ must be of the same type, e.g., all being "points you receive when asking a question in this website". They are individual observations of those points received. I can't say why you choose to study them jointly when they are of different types. It must be a mistake of your class.
– Alex
Oct 13, 2022 at 18:55
• @Alex Ok ! I think its more a misunderstanding from my side though. I thought that "explanatory variables are different" meant that they literally represent different things for example $x_{21}$ would have represented years of experience and $x_{31}$ a shot power. But if $Y_1, Y_2, Y_3$ are of the same type then $x_{21}$ and $x_{31}$ represent the same concept but with different numerical quantities. I think I got it Oct 13, 2022 at 19:04

You are correct, it wouldn’t make sense to have single model explaining completely unrelated things. $$(y_i, X_i)$$ are all of the same kind. For example $$y_i$$ is some economic indicator where $$i$$ is index for different countries and $$X_i$$ are the independent features for the countries (same for each country). Another example could be where $$y_i$$ is height of the child and $$x_i$$ height of the parent, where $$i$$ is an index for different parent-child pairs, as in the famous example by Sir Francis Galton.