OLS coefficient bounds

If $Y \sim X$ gives coefficient $\hat\beta$, what is the bound on the coefficient $\hat\theta$ from $X \sim Y$?

Solution

Note that

\[\begin{gathered} \hat\beta = \mathrm{Cov}(X,Y) / \mathrm{var}(X) \\ \hat\theta = \mathrm{Cov}(X,Y) / \mathrm{var}(Y) \end{gathered}\]

and so

\[\hat\beta\hat\theta = \mathrm{Cor}(X,Y)^2\]

Since correlation is bounded between $-1$ and $1$, we must have $\hat\beta\hat\theta$ is bounded between $0$ and $1$. Thus $\hat\theta \in \left[0, \hat\beta^{-1}\right]$.