hello. yes, it is a small part of a wider problem here (i'll try to be brief).

here in social-sciency land we have a regression-based method called 'mediation' where you have three variables (a predictor/independent variable X, a response/dependent variable Y and a mediator Z). the way it works if by first running the regression:

\(Y=b_{0} + b_{1}X\) and you look to see if \(b_{1}\) is significant

then you do other regressions (you predict Z from X, you predict Y from Z, nothing too important for this question).

what matters, however is that when then you run this regression:

\(Y=b_{0} + b_{1}X + b_{2}Z\) you need to see see how the coefficient \(b_{1}\) changes. if it becomes non-significant then you say Z "fully mediates" the relationship between X and Y (which rarely happens). if \(b_{1}\) is still significant but it's reduced (the most common case) then that means Z "partially mediates" X and Y.

we reviewed these concepts in class last tuesday and i was thinking to myself "well, it seems like in the most common case of partial mediation (i.e. \(b_{1}\) is still significant but smaller once Z is introduced in the regression equation) it would be useful to know the range of values \(b_{1}\) could have. that made me think about

Dragan's formulas for regression coefficients and the bounds that correlations impose each other to keep the correlation matrix as positive-definite. that's when i thought "what if i could find a way to provide a range of values that \(b_{1}\) can have when Z is introduced verus absent in the regression equation? which prompted my question.

but the formulas that Dragan posted have too much going on within them. i'm thinking there could always be a way that if something changes any potential range of values i could generate for \(b_{1}\) could be violated.