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 a general stationary processes possessing a spectral density with a dominant peak at some frequency ${\lambda }_0\in (0,\pi )$. We show that if the peak of the spectral density is sharp enough (in a way to be specified) then the AR(2) model, which best (in mean square sense) approximates the underlying process, correctly identifies the frequency ${\lambda }_0$. To investigate consistency properties of the AR(2) based estimator of ${\lambda }_0$, a near to pole framework is adopted. Triangular arrays of stationary stochastic processes are considered that possess a spectral density the peak of which at ${\lambda }_0$ becomes more pronounced as the sample size $n$ of the observed time series increases to infinity. It is shown in this set up, that the AR(2) based estimator achieves a rate of convergence which is larger than the parametric $n^{-1/2}$ rate and which can be arbitrarily close to $n^{-2/3}$, the best rate that can be achieved by this estimator.