say you have 3 variables X, Y and Z each one with some correlation [MATH]r_{xy}[/MATH], [MATH]r_{yz}[/MATH], [MATH]r_{xz}[/MATH]. we know from the formula of the determinant of the correlation matrix that if, for instance, [MATH]r_{xy}[/MATH] and [MATH]r_{xz}[/MATH] are fixed, then [MATH]r_{yz}[/MATH] must necessarily fall within the interval:

[MATH]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})}[/MATH]

so the question now becomes... if we consider the OLS multiple regression models [MATH]Y=b_{0}+b_{1}X[/MATH] and [MATH]Y=b_{0}+b_{1}X+b_{2}Z[/MATH], is there some way to calculate the range of values that [MATH]b_{1}[/MATH] can have when [MATH]Z[/MATH] gets introduced into the model? in general, the [MATH]b_{1}[/MATH] 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!