Chapter III - THE SPECTRAL REPRESENTATION AND ITS ESTIMATION

Abstract

This chapter presents two ways of introducing the spectral density function for stationary stochastic processes. First is a development based upon the Wold decomposition theorem and the autocorrelations of a time series and second shows how the spectral distribution function of a stationary time series can be derived from the representation of such a process in terms of random variables defined in the frequency domain rather than in the time domain. The chapter discusses the canonical factorization of the covariance generating function. The autocovariance generating function converges on the unit circle and in an annulus surrounding it. Stationary processes that have both a moving-average and an autoregressive representation are called invertible. One way to view the requirement of invertibility is as a condition for identifiability in the usual econometric sense. The chapter also discusses the spectral representation of a stationary time series. It is not necessary to assume stationarity for the development of the spectral representation but only covariance or weak stationarity. It is convenient to work with complex-valued time series until the last moment and then to make the necessary specialization.