Corr(X, Y) = 0.8 and Corr(Y, Z) = 0.8. What is the smallest value Corr(X, Z) can take, and what does the extreme configuration look like?

Corr(X, Y) = 0.8 and Corr(Y, Z) = 0.8. What is the smallest value Corr(X, Z) can take, and what does the extreme configuration look like?

Approach: The three-by-three correlation matrix must be positive semidefinite, so require its determinant to be non-negative and solve the resulting quadratic in the unknown correlation.

0.28. A correlation matrix must be positive semidefinite, and for three variables the binding determinant condition is 1 + 2 r_xy r_yz r_xz - r_xy^2 - r_yz^2 - r_xz^2 >= 0. With r_xy = r_yz = 0.8 this is a quadratic in r_xz with roots r_xy r_yz plus or minus sqrt((1 - r_xy^2)(1 - r_yz^2)) = 0.64 plus or minus 0.36, so r_xz lies in [0.28, 1]. Geometrically the variables are unit vectors whose angles obey a triangle inequality on angles: correlation 0.8 is an angle of 36.87 degrees, so X and Z are at most 73.74 degrees apart and cos(73.74) = 0.28. Two variables each tied strongly to a third cannot be close to independent.

Follow-up: If all three pairwise correlations are equal to r, what is the smallest r the matrix allows?

Key concepts: positive semidefinite, correlation matrix, determinant condition, triangle inequality on angles.