Admissibility intervals for linear correlation coefficients

The correlation coefficient is widely used to quantify the degree of association between two quantitative variables. By resorting to the geometric representation of the linear correlation coefficient, it is possible to calculate the upper and lower bounds of the correlation coefficient between two variables x ₁,x ₂ when the correlation coefficients with a third variable x ₃ are available. Implications in observational studies, where x ₃ could be a proxy of a target variable x ₂, whose direct measurement is too expensive or impractical, are discussed.