Inference for the Spectrum of a Stationary Process

Abstract

In this chapter we consider problems of statistical inference for time series based on frequency-domain properties of the series. The fundamental tool used is the periodogram, which is defined in Section 10.1 for any time series {x ₁,..., x _n }. Section 10.2 deals with statistical tests for the presence of “hidden periodicities” in the data. Several tests are discussed, corresponding to various different models and hypotheses which we may wish to test. Spectral analysis for stationary time series, and in particular the estimation of the spectral density, depends very heavily on the asymptotic distribution as n → ∞ of the periodogram ordinates of the series {X ₁,..., X _n }. The essential results are contained in Theorem 10.3.2. Under rather general conditions, the periodogram ordinates I _n (λ _i ) at any set of frequencies λ ₁,..., λ _m , 0 < λ ₁ < ⋯ < λ _m < π, are asymptotically independent exponential random variables with means 2πf(λ _i ), were f is the spectral density of {X _t }. Consequently the periodogram I _n is not a consistent estimator of 2πf. Consistent estimators can however be constructed by applying linear smoothing filters to the periodogram. The asymptotic behaviour of the resulting discrete spectral average estimators can be derived from the asymptotic behaviour of the periodogram as shown in Section 10.4. Lag-window estimators of the form ${\left( {2\pi } \right)^{ - 1}}\sum\nolimits_{\left| h \right| \leqslant r} {w\left( {h/r} \right)\hat \gamma \left( h \right)} {e^{ - ih\omega }}$, where w(x), −1 ≤ x ≤ 1, is a suitably chosen weight function, are also discussed in Section 10.4 and compared with discrete spectral average estimators. Approximate confidence intervals for the spectral density are given in Section 10.5. An alternative approach to spectral density estimation, based on fitting an AR MA model to the data and computing the spectral density of the fitted process, is discussed in Section 10.6. An important role in the development of spectral analysis has been played by the fast Fourier transform algorithm, which makes possible the rapid calculation of the periodogram for very large data sets. An introduction to the algorithm and its application to the computation of autocovariances is given in Section 10.7. The chapter concludes with a discussion of the asymptotic behaviour of the maximum likelihood estimators of the coefficients of an ARMA(p, q) process.