I have sample of three variables $X,Y,Z$. I estimated the equation using least squares:
$$ X = \alpha_1 Y + \alpha_2 Z.$$
I thought that if I estimate another equation: $$Y = \beta_1 X + \beta_2 Z,$$ the coefficients $\beta_1$ and $\beta_2$ should be equal to $\frac{1}{\alpha_1}$ and $-\frac{\alpha_2}{\alpha_1}$ because from first equation:
$$\frac{1}{\alpha_1}X = Y + \frac{\alpha_2}{\alpha_1}Z,$$
$$Y = \frac{1}{\alpha_1}X - \frac{\alpha_2}{\alpha_1}Z.$$
But when I made the second regression the coefficients were quite different from what I expected.
My question: Is my assumption even correct that from first equation we can derive coefficients of second equation or least squares doesn't work like that?
This is normal when there are errors since you do not minimize the same objective function.
$$SSQ_1= \sum_{i=1}^n \big[ \alpha_1 Y_i + \alpha_2 Z_i-X_i\big]^2$$ $$SSQ_2= \sum_{i=1}^n \big[\beta_1 X_i + \beta_2 Z_i-Y_i\big]^2$$
If you want to be "neutral", the solution could be the problem of the best fitting plane. Have a look here for the solution.