i feel like the answer to my question is a 'no' but i'll ask it anyway just to be absolutely sure.
say you have 3 variables X, Y and Z each one with some correlation \(r_{xy}\), \(r_{yz}\), \(r_{xz}\). we know from the formula of the determinant of the correlation matrix that if, for instance, \(r_{xy}\) and \(r_{xz}\) are fixed, then \(r_{yz}\) must necessarily fall within the interval:
\(r_{xy}r_{xz}-\sqrt{(1-r^{2}_{xy})(1-r^{2}_{xz})}\leq r_{yz}\leq r_{xy}r_{xz}+\sqrt{(1-r^{2}_{xy})(1-r^{2}_{xz})}\)
so the question now becomes... if we consider the OLS multiple regression models \(Y=b_{0}+b_{1}X\) and \(Y=b_{0}+b_{1}X+b_{2}Z\), is there some way to calculate the range of values that \(b_{1}\) can have when \(Z\) gets introduced into the model? in general, the \(b_{1}\) will not be the same in the first and in the second model. i was hoping maybe some function of maybe the correlations/covariances and variances of the constituting variables could give me a range of values...
thaaanks!
say you have 3 variables X, Y and Z each one with some correlation \(r_{xy}\), \(r_{yz}\), \(r_{xz}\). we know from the formula of the determinant of the correlation matrix that if, for instance, \(r_{xy}\) and \(r_{xz}\) are fixed, then \(r_{yz}\) must necessarily fall within the interval:
\(r_{xy}r_{xz}-\sqrt{(1-r^{2}_{xy})(1-r^{2}_{xz})}\leq r_{yz}\leq r_{xy}r_{xz}+\sqrt{(1-r^{2}_{xy})(1-r^{2}_{xz})}\)
so the question now becomes... if we consider the OLS multiple regression models \(Y=b_{0}+b_{1}X\) and \(Y=b_{0}+b_{1}X+b_{2}Z\), is there some way to calculate the range of values that \(b_{1}\) can have when \(Z\) gets introduced into the model? in general, the \(b_{1}\) will not be the same in the first and in the second model. i was hoping maybe some function of maybe the correlations/covariances and variances of the constituting variables could give me a range of values...
thaaanks!