We consider two important features of the historical US price data (1774–2015), namely the data’s persistence and cyclical structure. We first consider the persistence of the series and focus on standard long-memory models that incorporate a peak at the zero frequency. We examine different models with respect to the deterministic terms, including nonlinear deterministic trends of the Chebyshev form. Then, we investigate a more general model that includes both persistence and cyclicality of the series and, thus, includes two fractional integration parameters, one at the zero (long-run) frequency and the other at the nonzero (cyclical) frequency. We model the cyclical structure as a Gegenbauer process. This specification outperforms the standard long-memory specifications. We find that the order of integration at the zero frequency is about 0.5, and the one at the cyclical frequency is about 0.2 with cycles repeating approximately every 6 years, producing mean-reverting long-memory effects at both the zero and cyclical frequencies. Fitting the values to this model, however, we discover the presence of a break that, according to the methods employed, takes place at around 1940–1941. The results indicate the prevalence of the long-run or zero component with a much higher degree of persistence during the second post-1940–1941 subsample, suggesting important implications for monetary policy.