Least squares estimation question

Question

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?

Answer 1

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.