Bound a Missing Pairwise Correlation

Three standardized random variables have $$ \operatorname{Corr}(X_1,X_2)=0.8,\qquad \operatorname{Corr}(X_2,X_3)=0.8. $$ What can be concluded about (operatorname{Corr}(X_1,X_3))? Derive the full feasible interval and construct examples showing that the missing correlation is not uniquely determined. ### Constraints & Assumptions - Treat the three values as entries of a valid correlation matrix. - Do not assume a Markov chain, joint normality, or independence of residuals unless you state it as an added model. - Examples may be linear combinations of independent standardized variables. ### Clarifying Questions to Ask - Are we asked for an exact value or all values consistent with a positive-semidefinite correlation matrix? - Has any conditional-independence structure been specified? - Do the variables have finite, nonzero variance? ```hint Check the determinant A correlation matrix must be positive semidefinite. Apply that condition to the determinant of the three-by-three matrix. ``` ### What a Strong Answer Covers - Recognition that pairwise correlations do not obey transitivity. - The positive-semidefinite constraint and the resulting interval. - At least two valid constructions with different (X_1,X_3) correlations. - Clear separation of mathematical constraints from optional modeling assumptions. ### Follow-up Questions - What value results if (X_1\perp X_3\mid X_2) in a jointly Gaussian model? - How would sampling error affect a correlation matrix near the feasibility boundary? - How can an invalid empirical correlation matrix be projected to a valid one?