Just pondered over the following thought experiment for lunch. Suppose I am meant to perform an OLS for $Y=\beta X+\epsilon$, but when estimating $\hat\beta$ I accidentally use $X$ for $Y$ and $Y$ for $X$. What are the consequences of being "lazy" by rewriting the OLS as $X=(Y-\epsilon)/\beta=\gamma Y+\eta$ ($\gamma=1/\beta$ and $\eta=-\epsilon/\beta$), and instead of estimating $\gamma$, use $\hat\gamma=1/\hat\beta$?
My intuition so far: for the simple case of $X$, $Y$ being zero mean r.v.s., the right answer for minimising $\mathbb{E}[(Y-\beta X)^2]$ gives $\hat\beta=\frac{\rho\sigma_Y}{\sigma_X}$, and likewise for minimising $\mathbb{E}[(X-\gamma Y)^2]$ gives $\hat\gamma=\frac{\rho\sigma_X}{\sigma_Y}$. However, we accidentally claim that $\hat\beta=\frac{\rho\sigma_X}{\sigma_Y}$, and flipping it gives $\frac{\sigma_Y}{\rho\sigma_X}$. Since $\rho\in[-1,1]$, this leads to an overestimate when we apply our new model to a new dataset. I'm wondering if there is a deeper story to this and I am missing out on anything on my analysis? For example, if it is dependent on $\epsilon$/the distribution of the data? Thank you!
