Tests for Deterministic Versus Indeterministic Cycles

An indeterministic cycle can be represented as an infinite moving-average process and has a smooth peak in its spectrum. A deterministic cycle can be formulated in such a way that it is still stationary but its cyclical behaviour is characterized by a sudden jump in the spectral distribution function. This paper shows how both processes can be nested within the same model and how, when the cycle is buried in noise, the null hypothesis of a deterministic cycle can be tested. The preferred tests are based on the residuals from regressing the observations on sine and cosine functions of the frequency of interest. They are derived as point optimal and locally best invariant tests and can be extended to regression models which include other explanatory variables. The asymptotic distribution of the locally best invariant test, which is a one-sided score test, is shown to be Cramer–von Mises with two degrees of freedom. Modifications for dealing with serially correlated noise are discussed and the methods are illustrated with real data.