I recently received an exam question where I was asked to compare the regression coefficients of 2 linear regression models (see below) as well as their predicted values. The dependent variables var1 and var2 are highly correlated with each other.
y ~ var1 + var2
w1 = var1 + var2
w2 = var1 - var2
w ~ w1 + w2
Can I conclude that in both cases both independent variables are very highly correlated with each other which makes the regression coefficients very unstable? Also, both models will have different regression coefficients that are not statistically significant.
As an additional question I was asked what I conclude regarding the predicted value for y and w? Since both independent variables are highly correlated, and both models are based on the same independent variables I would assume that the predicted value for both models will be the same, but maybe I’m wrong?