Regression model result interpretation I am trying to learn few things about generalized least square regression modeling.
Here are the models that I am using.,
X1 ~ Y1 + Y2,
X2 ~ Y1 + Y4,
All predictor variables, Y1, Y2 and Y4 are significantly related with the response variables, but when I took the difference of two response variables and the difference of two predictor variables, something like this:
X3 = X1 - X2,
Y5 = Y2 - Y4,
and use this model
X3 ~ Y1 + Y5
Now, Y5 is not significant:
I am confuse about it. As, Y5 is just a difference of Y2 and Y3 and X3 is the difference of X1 and X2. 
Can someone guide me? 
 A: Writing theoretical models in man-machine communication language, and not in mathematical form, has various undesirable effects. In your case, it puts out of sight the object of the whole endeavor- the unknown coefficients which are to be estimated.
You have two equations
$$X_1 = a_0 + a_1Y_1 + a_2Y_2 + u_1 \tag{1}$$
$$X_2 = b_0 + b_1Y_1 + b_2Y_4 + u_2 \tag{2}$$
Assume that these are correct specifications and that they satisfy the usual assumptions of least-squares regression. Then the correct specification of the difference of the two variables is
$$X_3 \equiv X_1 - X_2 = (a_0-b_0) + (a_1-b_1)Y_1 + a_2Y_2 -b_2Y_4 + (u_1-u_2) \tag{3}$$
Instead, you specified
$$X_3 = c_0 + c_1Y_1 + c_2(Y_2-Y_4) + u_3 \tag{4}$$
We see that we can map
$$c_0 = a_0-b_0,\;\;\; c_1=a_1-b_1,\;\;\; u_3=u_1-u_2$$
and we are ok on these aspects. But instead of the correct $a_2Y_2 -b_2Y_4$ we specified $c_2(Y_2-Y_4)$. Except from the special case where $a_2=b_2$, in all other cases, we postulated in $(4)$ a linear relationship between the difference of $X_1-X_2$ and the difference $Y_2-Y_4$ while the correct specification is that the difference $X_1-X_2$ depends on the term $a_2Y_2 -b_2Y_4$. So equation $(4)$ contains most likely a misspecification.  
The fact that, on account of equations $(1)$ and $(2)$, the difference $X_3 = X_1-X_2$ covaries with $a_2Y_2 -b_2Y_4$, does not guarantee that it will also co-vary with  $Y_2 -Y_4$ and so it does not guarantee that $c_2$ will come out statistically significant.  
Reversely, the fact that $c_2$ comes out statistically insignificant, indicates that $a_2 \neq b_2$. And this is because, if $a_2=b_2 = \gamma \neq 0$ (different than zero because under the assumption of initial correct specification, $a_2$ and $b_2$ are non-zero) then $\gamma = c_2$ and so $c_2$ should not come out statistically insignificant. So $c_2 = 0 \Rightarrow a_2 \neq b_2$. Does the corresponding estimates $\hat a_2$ and $\hat b_2$ conform with this (informally speaking, are they "different enough")?
