Simply, Lets us say that we know $m_{xy}$ from the linear equation $y= m_{xy}*x + c$; And now we want to calculate $m_{yx}$ because we want to estimate the effect of x on y i.e $x = m_{yx} * y + c$ given $m_{xy}$; Can we do that?
To Elaborate, This is what I have tried; Let us say X and Y are both standardized with standard deviation of one and mean of zero then by theory $m_{xy} = m_{yx}$; these should be equal..
For this example,
let say $m_{xy}$ =0.7, so when x=1
$$y = m_{xy}*x+c$$
$$y = m_{xy}*1+0$$
$$y = 0.7 *1 =0.7$$
Now from the equation above, when y=0.7; then x should be equal to 1 . Substituting these values for the linear equation y on x, which is just like finding an inverse
$$x = m_{yx}*y+c $$ $$x = m_{yx}*0.7+0 $$ $$ m_{yx} = 0.7/1 =1.428$$
And I get $m_{xy} = 0.7$ and $m_{yx} = 1.428 $ these are clearly not equal which does not abide by the theory where these should be equal.
Also I tried fitting a linear regression here using OLS which gives me the right answer $m_{yx} = 0.7 $. But when I use this inverse function as demonstrated above to calculate $m_{yx}$ I get a different answer. I know these two methods are totally different but I cannot picture their difference yet. I think they both should satisfy and should have yield the same estimate.
Am I missing something here?? Thanks in advance!!