"Lets say we have a linear regression function y=a+bx +e, where b is the coefficient, a is the intercept, and e is the error term. However, after more research, you suspect that the equation is really x = a+by +e. How does the coefficient b change when you're testing the effect of y on x, as opposed to x on y?"

I'm sure that I'm over complicating this question or forgetting something basic, but can I assume that the coefficient b changes only by the value of y, on average, rather than the value of x?

Thanks so much for the help! It's been a while since I've taken anything related to stats

So for your first set, y = a + bx + e, y changes linearly as a function of x with a direct slope of b.

When you rearrange these variables to now account for x in terms of y, you can rewrite it as x = a + by + e, but this doesn't hold because a b and e are not the same a b and e that are in y = a + bx + e.

As an example:

y = 1 + 2x + 0.01 where a = 1, b = 2, e = 0.01

to rewrite this equation in terms of x = a + by + e, we shift over a and e to the y side, and then divide by b for both sides.

The equation now becomes: x = 1/2 (y - 1 - 0.01) or x = 1/2*y - 1/2 - 0.005

For this second case, a is now -1/2, b is now 1/2, and e is now -0.005.

This is how, assuming we aren't writing a brand new equation for the new function, the coefficient b changes. It changed from 2 to 1/2, effectively inversing itself, or taking 1 over the previous value to get the new value.

Do note that a and e will also always be scaled by b and will have their signs flipped in this type of equation.

Pm me if you need more clarification or if I didn't do exactly what you were looking for.