Econometric Time Series Long-Run Linkages and Parameter Instability in the Gold–Silver Relationship, 2010–2025 #gold_silver_cointegration_fractional_integration_structural_breaks_PELT_algorithm_Bai–Perron_test_fractional_cointegration_FCVAR This paper examines long-run linkages and possible instabilities in the gold–silver price relationship using daily futures prices over the period from 4 January 2010 to 28 November 2025. The empirical analysis includes unit-root and cointegration tests as well as endogenous structural break tests, namely the Pruned Exact Linear Time (PELT) algorithm applied to the Engle–Granger residuals and the Bai–Perron endogenous break test, both detecting a break in late 2017. Standard cointegration tests fail to detect a stable long-run equilibrium over the full sample and the pre-break subsample, while one is found in the post-break subsample. Further, the Local Whittle fractional integration method provides evidence of a high degree of persistence consistent with long-memory dynamics. The estimation of a Fractionally Cointegrated VAR (FCVAR) model corroborates the previous findings: although full-sample evidence for cointegration is limited, a stable and economically meaningful long-run relationship between gold and silver emerges clearly in the post-break period. The results are shown to be robust across frequencies. (author: Luis Alberiko Gil-Alana)