Detecting Periodicity of a General Stationary Time Series via AR(2)‐Model Fitting

ABSTRACT Estimating the periodicity of a stationary time series via fitting a second‐order stationary autoregressive (AR(2)) model has been initiated by the seminal paper of Yule (1927). We investigate properties of this procedure when applied to general stationary processes possessing a spectral density with a dominant peak at some unknown frequency . For this, a general class of stationary processes is considered with spectral densities having an arbitrary sharp peak. It is shown within this class, that if the peak of the spectral density is strong enough (in a sense to be specified), then the AR(2) model, which best (in mean square sense) approximates the underlying process, correctly identifies the frequency . To investigate consistency properties of the corresponding AR(2) based estimator of , a near‐to‐pole asymptotic framework is adopted. Triangular arrays of stationary stochastic processes are considered that possess a spectral density the peak of which at frequency becomes more pronounced as the sample size of the observed time series increases to infinity. It is then shown that, depending on the rate at which the sharpness of the spectral density at gets closer to that of a pole, the AR(2) based estimator achieves a rate of convergence which is larger than the parametric rate and can be arbitrarily close to , the best rate that can be achieved by this estimator.