Estimation of Random Cycles in Persistent Time Series

A number of economic, financial, and climatic time series exhibit persistent cycles which are characterized by time-dependence patterns and peaks in the spectrum. In this article, we introduce a class of semiparametric cyclical–memory processes which enable the modeling of random cyclical patterns in stationary and non-stationary time series. We develop a theoretical background and asymptotic estimation theory for the frequency of a cycle represented by the location of a peak in the spectrum. The estimation procedure is easy to implement and allows for the construction of narrow confidence intervals around the location point. Monte Carlo simulations confirm the good finite sample performance of our estimator. We illustrate our method with three empirical applications. We uncover (quasi-)periodic cycles in macroeconomic series, both nominal and real (U.S. nominal GDP and real industrial production), and CO 2 CO₂ upper C upper O 2 concentration levels.