I have two one dimensional dataset $X$ and $Y$.
I run regression and obtained $A$ from $Y = AX$. And another regression and obtain $B$ from $X = BY$. What's the relationship between $A$ and $B$? Is it always $A = 1/B$? My computer simulation suggests otherwise...
In simple regression, $y=\beta_0+\beta_1 x+\epsilon$, $\hat{\beta}_1=r\frac{s_y}{s_x}$.
By symmetry, for $x=\alpha_0+\alpha_1 y+\eta$, $\hat{\alpha}_1=r\frac{s_x}{s_y}$.
Clearly, then, ${\hat{\alpha_1}}= \frac{r^2}{\hat{\beta_1}}$, which is always $\leq \frac{1}{\hat{\beta_1}}$.
See the additional information in gung's excellent answer here. It doesn't directly answer this question, but it provides you with a lot of the tools and intuition that's useful for taking the extra small steps from that answer.
