# Linear regression and interpretation of random variables

in my statistics course, it was mentioned that for the Ordinary least squares method (OLS) taking variables $X_i$ and $Y_i$ as the response variable, the random variable is solely $Y_i$.

However for the Orthogonal regression method (ORM) both the $X_i$ and $Y_i$ are considered to be random variables.

From my understanding in the OLS we optimize the regression based on individually picked values, so for each $x_i$. However, in the ORM the optimization is not made for some fixed $x_i$ value. Is this the correct reason, or is there something else?

It's not true that, with OLS, only the dependent variable, $$Y$$, is random. In fact, both can be random variables, however OLS is centered on minimizing the mean squared deviation between $$Y$$ and $$X\beta$$, conditioned on the variable $$X$$. That is, if your equation is $$Y = X\beta + e$$ where $$e$$ is a vector of IID components, then the least squares problem is to find the vector $$\beta$$ such that $$\mathbb{E}[(Y- X\beta)^2\mid X]$$ is minimized.
In this sense, we can often "treat" the random variable $$X$$ as being "non-random", since we can pull it out of the operator $$\mathbb{E}[ \cdot \mid X]$$, however it still can be a random variable.
ORM has to do with dealing with cases where both $$X$$ and $$Y$$ contain measurment error. That is, when $$X = X + \epsilon$$ and $$Y = y + \mu$$ where $$x,y$$ are unknown fixed constants and $$\epsilon, \mu$$ are independent random variables.
• My contribution is to add a little correction on Digio contribution. First of all, thank for your answer, it is very clear. It is not a particular comment but the fact that it is not $X$ we are going to find but the vector $\beta$ who minimize the quantity $$\mathbb{E} [ (Y - X \beta )^ 2 | X ]$$ Commented Dec 16, 2023 at 3:41
• @ThegoatKonan Thank you for noticing that. I have fixed this evident typographical error by changing "$X$" to "$\beta.$"