Common mistake 1 with Spectral analysis

Forgetting to detrend and then interpreting the trend as the dominant "cycle".

Common mistake 2 with Spectral analysis

Reading every small bump as a real cycle. The periodogram is extremely noisy.

Common mistake 3 with Spectral analysis

Reading the largest peak as significant because it is the largest. The maximum of many noisy frequencies is large by construction, and testing the peak you found is not a valid test.

Common mistake 4 with Spectral analysis

Applying it to a short series and reporting a period longer than half the sample. Nothing about a cycle can be established from one repetition.

EconStatsCraft › Spectral analysis

Spectral analysis

Which cycle lengths dominate the series? The periodogram and the smoothed Welch spectrum answer that in the frequency domain.

What it needs

Series

The intuition

Any stationary series can be written as a sum of sine waves of different frequencies. The spectrum tells you how much of the variance sits at each frequency - the frequency-domain twin of the autocorrelation function, carrying exactly the same information in a form that makes cycle LENGTH obvious rather than implicit.

When to use it

When you suspect periodicity and want to find its length rather than assume it - seasonality of unknown period, or business-cycle frequencies.

Where the estimator comes from

The spectral density is the Fourier transform of the autocovariance function, f(\omega)=\frac{1}{2\pi}\sum_k \gamma_k e^{-i\omega k}. The periodogram is its sample analogue, I(\omega)=\frac{1}{2\pi n}\bigl|\sum_t y_t e^{-i\omega t}\bigr|^2. The periodogram is an unbiased but INCONSISTENT estimator - its variance does not shrink as n grows - which is why it must be smoothed, and Welch does that by averaging over overlapping segments.

The model

f(\omega)=\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}\gamma_k e^{-i\omega k}\qquad I(\omega)=\frac{1}{2\pi n}\left|\sum_{t=1}^{n}y_t e^{-i\omega t}\right|^{2}

What it assumes

The series is stationary - otherwise the trend dominates every frequency.
The cycle is stable in frequency across the sample. One that lengthens over time smears across neighbouring frequencies and can vanish from the periodogram entirely.
The sampling frequency is at least twice the frequency of interest. Anything faster than the Nyquist frequency is not merely missed - it is aliased, appearing as a slower cycle that is not there.
The series contains several repetitions of the cycle. One or two cannot be distinguished from a trend.

Statistical properties

The area under the spectrum equals the variance of the series.
White noise has a flat spectrum; that is what "white" means.
Power concentrated near frequency zero means a trend or unit root, not a cycle.
The highest resolvable frequency is the Nyquist limit, one cycle per two observations - monthly data can never reveal a weekly cycle.
Welch trades frequency resolution for lower variance; more segments give a smoother but blunter picture.

How to read the output

Read the "period" column, not the frequency: a peak at period 12 in monthly data is annual seasonality.
Power rising steeply towards frequency zero means a trend or a unit root, not a cycle.
A flat spectrum is white noise.
The Welch estimate is the readable one; the raw periodogram is noisy by construction.
The periodogram at each frequency has about two degrees of freedom regardless of sample size, so more data buys more frequencies rather than more precision at each. Smoothing - the Welch estimate shown beside it - is what buys precision, at the cost of resolution.
Read the period column rather than the frequency: a peak at 0.0833 means nothing until it is 12 observations per cycle.
Total power equals the variance of the series, so the share column tells you what fraction of the variation each band accounts for.

A worked reading

The strongest peak sits at frequency 0.083, i.e. a period of about 12 observations. On monthly data that is an annual cycle - so a seasonal ARIMA with S = 12, or a seasonal decomposition, is warranted.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Huge peak at frequency ≈ 0	Trend, not a cycle	Detrend or difference and re-run
Peak at 1/4 in quarterly data	Annual seasonality	Seasonally adjust or add seasonal terms
Flat spectrum	White noise	Nothing to forecast from its own history
Many small spikes	Periodogram noise	Read the Welch estimate instead

Common mistakes

Forgetting to detrend and then interpreting the trend as the dominant "cycle".
Reading every small bump as a real cycle. The periodogram is extremely noisy.
Reading the largest peak as significant because it is the largest. The maximum of many noisy frequencies is large by construction, and testing the peak you found is not a valid test.
Applying it to a short series and reporting a period longer than half the sample. Nothing about a cycle can be established from one repetition.

Where to read this properly

Hamilton (1994), Time Series Analysis, Princeton University Press, chapter 6
Welch (1967)
Priestley (1981), Spectral Analysis and Time Series

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Spectral analysis

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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