12.2 The Periodogram
Definition 12.2.1. Given data \(x_1,\dots ,x_n\), the periodogram at the Fourier frequencies \(\omega _j = 2\pi j/n\) is \[I(\omega _j) = \frac {1}{2\pi n}\left |\sum _{t=1}^{n} x_t e^{-i\omega _j t}\right |^{2}.\]
Remark. The periodogram is the natural estimate of the spectral density and it is a bad one. It is asymptotically unbiased, but its variance does not fall as \(n\) grows: at each frequency \(I(\omega _j)\) is approximately \(f(\omega _j)\chi ^{2}_{2}/2\), whose variance is \(f(\omega _j)^{2}\) however long the series. Lengthening the record adds more frequencies rather than improving any one of them.
The cure is smoothing — averaging the periodogram over a window of neighbouring frequencies. Averaging \(m\) approximately independent ordinates cuts the variance by a factor \(m\) at the cost of blurring detail across the window, which is the usual bias-variance trade and is chosen by how sharp the features of interest are expected to be.
Note. This is worth stating plainly because it is counter-intuitive: an estimator can be consistent for the whole function in an integrated sense while being inconsistent at every single point. The periodogram is the standard example, and it is why raw periodograms look so ragged.
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