12.1 The Spectral Density
Definition 12.1.1. For a stationary process with autocovariance \(\gamma (\tau )\) satisfying \(\sum _{\tau }\left |\gamma (\tau )\right |<\infty \), the spectral density is \[f(\omega ) = \frac {1}{2\pi }\sum _{\tau =-\infty }^{\infty }\gamma (\tau )\,e^{-i\omega \tau }, \qquad -\pi \leq \omega \leq \pi .\]
Result 12.1.2. The autocovariance is recovered by inversion, \[\gamma (\tau ) = \int _{-\pi }^{\pi } f(\omega )\,e^{i\omega \tau }\,d\omega ,\] and in particular \[\gamma (0) = \Var (X_t) = \int _{-\pi }^{\pi } f(\omega )\,d\omega .\]
Note. The last line is the interpretation. The total variance of the series is distributed across frequencies, and \(f(\omega )\,d\omega \) is the share contributed by cycles near frequency \(\omega \). A peak in the spectrum is a frequency band doing a disproportionate amount of the work — which is exactly what a seasonal effect looks like, and it is visible at a glance where the autocorrelation function would show only a slow ripple.
Example 12.1.3. Find the spectral density of white noise, and of the AR\((1)\) process \(X_t = \phi X_{t-1} + \varepsilon _t\).
Solution.
White noise
Here \(\gamma (0)=\sigma ^{2}\) and \(\gamma (\tau )=0\) otherwise, so only one term survives: \[f(\omega ) = \frac {\sigma ^{2}}{2\pi },\] constant in \(\omega \). Every frequency contributes equally — which is what the name means, by analogy with white light.
AR(1)
For the AR\((1)\), \(\gamma (\tau ) = \dfrac {\sigma ^{2}\phi ^{|\tau |}}{1-\phi ^{2}}\). Summing two geometric series, \[f(\omega ) = \frac {\sigma ^{2}}{2\pi }\cdot \frac {1}{\left |1-\phi e^{-i\omega }\right |^{2}} = \frac {\sigma ^{2}}{2\pi \left (1 - 2\phi \cos \omega + \phi ^{2}\right )} .\] For \(\phi >0\) this is largest at \(\omega =0\): the series is dominated by slow movement, which is what positive autocorrelation means. For \(\phi <0\) the peak moves to \(\omega =\pi \), the fastest representable oscillation — alternation from one observation to the next.
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