0.1 Periodogram and Spectral Estimation
The spectral density \(f(u)\) of a stationary process describes how variance is distributed across frequencies \(u\). Given a finite sample \(x_1,\dots,x_n\), we estimate \(f\) empirically.
For frequencies \(u_k = 2\pi k/n\) (Fourier frequencies), the periodogram is \[ I_n(u_k) = \frac{1}{n}\left|\sum_{t=1}^n (x_t - \bar x)e^{-iu_kt}\right|^2 = \sum_{h=-(n-1)}^{n-1} \hat\gamma(h)e^{-iu_kh}. \] The periodogram is the discrete Fourier transform of the sample ACVF, and is the natural (though inconsistent) estimator of \(2\pi f(u)\).
Inconsistency and smoothing. \(I_n(u_k)\) has asymptotic variance that does not vanish as \(n\to\infty\) (it is asymptotically \(\chi^2_2\)-distributed at each frequency, independent across Fourier frequencies), so it is not a consistent estimator of the spectral density. Consistent estimators smooth the periodogram across neighboring frequencies: - Discrete spectral averaging / kernel smoothing: \(\hat f(u) = \sum_j W_n(j)I_n(u+\lambda_j)\) for a weight kernel \(W_n\), - Lag-window estimators: truncating/tapering the ACVF sum with a window \(w(h/M)\) before transforming, - Multitaper methods (Thomson): averaging periodograms computed with multiple orthogonal (Slepian/DPSS) tapers to reduce variance without excessive bias.
Uses. Spectral/periodogram methods detect hidden periodicities (peaks at the corresponding frequency), test for white noise (flat periodogram), and underlie frequency-domain estimation of \(ARMA\) models (Whittle likelihood). They complement, rather than replace, Wavelet Analysis, which localizes frequency content in time for non-stationary series.