Chapter 09

Hard

Spectral Analysis and Linear Filters

00 · Symbol Glossary

$f(\omega)$f of omega — spectral density

The frequency-domain partner of the autocovariance $\gamma(h)$ . On $(-\pi,\pi]$ , $f(\omega)\,d\omega$ is the contribution of frequencies near $\omega$ to the variance of the series.

$\gamma(h)$gamma of h — autocovariance

$\gamma(h)=\mathrm{Cov}(X_t,X_{t-h})$ for a weakly stationary series. Wiener–Khinchin says $\gamma$ and $f$ are Fourier duals.

$I(\omega)$I of omega — periodogram

A raw sample estimate of $f(\omega)$ from one stretch of data. Noisy; useful after smoothing, or as a diagnostic that a series is not white.

$\psi(e^{-i\omega})$filter gain at frequency omega

If $Y_t=\psi(B)X_t$ , the output spectrum is $\lvert\psi(e^{-i\omega})\rvert^2 f_X(\omega)$ . Band-pass and difference filters are statements about this gain.

01 · Autocovariance as a Spectrum

Weak stationarity packages all second-order dependence into $\gamma(h)$ . The spectral density unpacks that same information by frequency: cycles of period $2\pi/\omega$ rather than lags of length $h$ . This is not a second course in Fourier analysis. It is the ACF, rewritten so that “business-cycle band” and “high-frequency noise” become visible.

Theorem — Wiener–Khinchin (discrete time)

If $\{X_t\}$ is weakly stationary and $\sum_h\lvert\gamma(h)\rvert<\infty$ , there is a continuous spectral density

f(\omega)=\frac{1}{2\pi}\sum_{h=-\infty}^{\infty}\gamma(h)e^{-i\omega h}, \qquad \omega\in[-\pi,\pi]

and the inversion

\gamma(h)=\int_{-\pi}^{\pi} e^{i\omega h} f(\omega)\,d\omega

In particular $\gamma(0)=\mathrm{Var}(X_t)=\int_{-\pi}^{\pi} f(\omega)\,d\omega$ : variance is an integral of the spectrum.

The $1/(2\pi)$ convention is the one used in Hamilton and Brockwell–Davis. Other books fold $2\pi$ into $f$ ; do not mix formulas across conventions.

02 · White Noise, AR(1), and Peaks

Example — White noise is flat

If $\varepsilon_t\sim\mathrm{WN}(0,\sigma^2)$ , then $\gamma(0)=\sigma^2$ and $\gamma(h)=0$ for $h\neq 0$ , so

f_{\varepsilon}(\omega)=\frac{\sigma^2}{2\pi}

Every frequency contributes equally. A periodogram that is roughly level is the frequency-domain version of “the ACF is a spike at $0$ .”

Formula — Causal AR$(1)$ spectrum

For $X_t=\phi X_{t-1}+\varepsilon_t$ with $\lvert\phi\rvert<1$ and $\varepsilon_t\sim\mathrm{WN}(0,\sigma^2)$ ,

f_X(\omega)=\frac{\sigma^2}{2\pi}\cdot\frac{1}{\lvert 1-\phi e^{-i\omega}\rvert^2}=\frac{\sigma^2}{2\pi}\cdot\frac{1}{1+\phi^2-2\phi\cos\omega}

If $\phi>0$ , $f$ is largest at $\omega=0$ (low-frequency / persistent). If $\phi<0$ , the peak is at $\omega=\pi$ (period- $2$ oscillation). Persistence in the ACF is a low-frequency pile-up in $f$ .

A seasonal monthly series with a strong twelve-month cycle shows a peak near $\omega=2\pi/12$ . GARCH clustering is not a mean-spectrum story: it lives in the squares. Do not hunt for a spectral peak to “explain” volatility clustering.

03 · The Periodogram

Definition — Periodogram

For a sample $X_1,\ldots,X_T$ (mean-subtracted),

I(\omega)=\frac{1}{2\pi T}\left\lvert\sum_{t=1}^{T} X_t e^{-i\omega t}\right\rvert^2

At Fourier frequencies $\omega_j=2\pi j/T$ , $I(\omega_j)$ is (approximately) an unbiased but inconsistent estimator of $f(\omega_j)$ : variance does not fall with $T$ because each ordinate behaves like a noisy draw (for Gaussian series, roughly exponential at a fixed $\omega$ ), not an average that shrinks as $T$ grows. Smooth neighboring ordinates (a spectral window) to get a stable estimate of $f(\omega)$ .

❌ Treating one periodogram spike as a proven cycle

A single large $I(\omega_j)$ can be sampling noise, a leftover calendar effect, or leakage from a nearby frequency.

Why it breaks: the periodogram is asymptotically exponential (for Gaussian series, at a fixed $\omega\neq 0$ ), so the maximum over many frequencies is stochastically large even under white noise.

Consequence: prewhiten, smooth, and ask whether the peak survives a holdout or a different sample. Do not name a “cycle” from one noisy plot.

04 · Linear Filters

If $Y_t=\sum_{j}\psi_j X_{t-j}=\psi(B)X_t$ with absolutely summable weights, the spectra multiply by the squared gain:

f_Y(\omega)=\lvert\psi(e^{-i\omega})\rvert^2 f_X(\omega)

This is why differencing kills a unit-root pile-up at $\omega=0$ : $\psi(z)=1-z$ has $\lvert 1-e^{-i\omega}\rvert^2=2-2\cos\omega$ , which is $0$ at $\omega=0$ and large at high frequency. Overdifferencing an already stationary series therefore inflates high-frequency noise — the frequency-domain version of the MA unit-root warning in Chapter 04.

A band-pass filter is a $\psi$ whose gain is near $1$ on a chosen interval of $\omega$ and near $0$ elsewhere (business-cycle band, seasonal band). Moving averages are low-pass filters: they damp high $\omega$ . The Hodrick–Prescott filter is a popular high-pass for extracting a cycle; it is an ad hoc filter with a chosen smoothing constant, not a parametric time-series model whose spectrum you would estimate from ARMA/GARCH parameters.

What this chapter does not do

Fourier inversion proofs, continuous-time spectral measures, or a full treatment of leakage, tapering, and multitaper estimates. Those belong in a time-series / signal-processing sequel. The goal here is to read $\gamma(h)$ as $f(\omega)$ and to see a filter as a gain.

05 · Exercises

EXERCISE 9.1

Only $\gamma(0)$ is nonzero; plug into Wiener–Khinchin.

$f(\omega)=\gamma(0)/(2\pi)=\sigma^2/(2\pi)$ for all $\omega$ . Flat spectrum: white noise has no preferred frequency.

Using Wiener–Khinchin, compute $f(\omega)$ for $\mathrm{WN}(0,\sigma^2)$ .

EXERCISE 9.2

The AR $(1)$ formula is largest at $\omega=0$ when $\phi>0$ .

A peak at low frequency: $\phi=0.9$ puts most variance in long swings (slow ACF decay). That is persistence, not a business-cycle “period” unless you impose a band-pass on top.

For causal AR $(1)$ with $\phi=0.9$ , where is $f(\omega)$ largest, and what does that say about the series?

EXERCISE 9.3

$f_Y(\omega)=\lvert 1-e^{-i\omega}\rvert^2 f_X(\omega)$ and $\lvert 1-e^{-i\omega}\rvert^2=0$ at $\omega=0$ .

Differencing annihilates $\omega=0$ . If $X$ already had a spectral peak at zero (unit root or highly persistent AR), $\Delta X$ removes it. If $X$ was already WN, $\Delta X$ is MA $(1)$ with a hole at $\omega=0$ and extra high-frequency power — overdifferencing.

Let $Y_t=\Delta X_t$ . What does the filter gain do at $\omega=0$ ? What goes wrong if $X$ was already $I(0)$ white noise?

06 · Chapter Summary

Object	Role
$f(\omega)$	Fourier dual of $\gamma(h)$ ; variance by frequency
WN spectrum	flat, $\sigma^2/(2\pi)$
AR $(1)$	low-frequency peak if $\phi>0$
Periodogram	raw, inconsistent estimate of $f$ ; smooth it
Filter gain	$f_Y=\lvert\psi(e^{-i\omega})\rvert^2 f_X$
Differencing	high-pass that kills $\omega=0$

Next: Chapter 10 — State-Space Models and the Kalman Filter, when the object you care about (a latent factor, a time-varying $\beta$ ) is not observed directly.