Equations from two simple linear regression models defining the same straight line in the plane

Consider the two simple linear regression models:

$$Y = \alpha_0 + \alpha_1X + \epsilon(0, \sigma^2)$$ and $$X = \beta_0 + \beta_1Y + \epsilon(0, \sigma^2)$$.

Let $$(\hat{\alpha_0}, \hat{\alpha_1})$$ and $$(\hat{\beta_0}, \hat{\beta_1})$$ be the LSE of the regression coefficients in these models from the same set of observations $$\{(x_i, y_i) : 1 \leq i \leq n\}$$. Do the equations $$y = \hat{\alpha_0} + \hat{\alpha_1}x$$ and $$x = \hat{\beta_0} + \hat{\beta_1}y$$ define the same straight line in the plane ?

Here are my thoughts so far:

I was able to show that $$\hat{\alpha_1} = \frac{\sum_{i = 1}^n x_iy_i - n\bar{x}\bar{y}}{\sum_{i = 1}^n x_i^2 - n\bar{x}^2}$$ and $$\hat{\beta_1} = \frac{\sum_{i = 1}^n x_iy_i - n\bar{x}\bar{y}}{\sum_{i = 1}^n y_i^2 - n\bar{y}^2}$$. From here, I was also able to show that $$\hat{\alpha_1} \cdot \hat{\beta_1} \leq 1$$. Is there some reason that equality must hold here ? That is, $$\hat{\alpha_1} \cdot \hat{\beta_1} = 1$$ ? If so, how can I proceed from there to show that the two given equations define the same straight line in the plane ? Is the only route to take to show that $$\hat{\alpha_0} = \hat{\beta_0}$$ and $$\hat{\alpha_1} = \hat{\beta_1}$$ ?

Thanks for your help. I really appreciate it. I'm relatively new to this subject.

1 Answer

No, they don't. Just run one, and invert the equation, to see that this is so. The reason is that if you run the $$y=mx+b$$ equation, you're minimizing the sums of squares of $$y$$ distances, whereas if you run the $$x=my+b$$, you're minimizing the sums of squares of $$x$$ distances. Now if you minimized the right-angle distances from points to the best-fit line, you would get the same equation either way.