Yule (1927) On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers

Summary

The founding paper of the autoregressive (AR) model. Yule asks why Wolfer's sunspot series looks smoothly periodic yet continually shifts in amplitude and phase, and argues it should be modeled not as a fixed sinusoid plus measurement error but as a disturbed periodic movement — "a pendulum subjected to successive small random impulses." He fits a second-order linear regression / difference equation relating $u_x$ to $u_{x-1}$ and $u_{x-2}$ (i.e. an AR(2)), whose solution is a damped harmonic function driven by random disturbances. The paper's lasting conceptual contribution is the sharp distinction between superposed fluctuations (noise added to observations) and disturbances (random shocks fed into the dynamics and propagated forward).

Key Claims

• Two ways randomness enters a periodic series. Superposed fluctuations are errors added to the observed ordinates of an otherwise deterministic harmonic curve — they blur the graph but leave a fixed underlying period. Disturbances are random impulses that act on the system itself (like knocks to a pendulum), continually changing amplitude and phase and producing unpredictability that grows with time. Sunspots behave like the latter.
• The autoregressive difference equation. Yule fits, by least squares, a linear relation predicting $u_x$ from its own recent past ($u_{x-1}, u_{x-2}$) — a second-order stochastic difference equation. This is the AR(2) model; the deterministic part corresponds to a difference equation whose characteristic roots give a harmonic solution, and the residuals are the estimated "disturbances."
• Damped harmonic solution. Solving the fitted regression as a finite difference equation yields a rapidly damped harmonic function — so a stationary AR process sustains oscillation only because the disturbances keep re-exciting it, not because of an undamped periodic force.
• No spurious secondary period. Extending the equation to test for a second harmonic, $u_x$ is not better predicted by $u_{x-1}\dots u_{x-4}$ than by $u_{x-1}, u_{x-2}$ alone — evidence against a secondary period and in favor of the disturbance interpretation.
• Periodogram analysis can mislead. For genuinely disturbed series, periodogram/harmonic analysis gives results with a large margin of error (unstable with few cycles, band-broadening with many) — the wrong tool when the periodicity is stochastically maintained rather than fixed.

Concepts Introduced or Extended

• Autoregressive Model

Entities Mentioned

• G. Udny Yule

Quotes

"The sunspot numbers ... should be regarded as analogous to the data that would be given by observations of a disturbed periodic movement, such as that of a pendulum subjected to successive small random impulses."

"The result also suggests the existence of disturbances (as distinct from superposed fluctuations), since only disturbances can give the required element of unpredictability rapidly increasing with the time."

My Take

This is the paper time-series econometrics grows out of, and its central idea is still the one students most need to internalize: a stochastic process can look periodic without containing any fixed periodic component — the oscillation is a damped resonance of a difference equation that random shocks keep reigniting. The superposed-fluctuations vs. disturbances distinction is exactly the modern difference between measurement error on the observation equation and innovations in the state/transition equation, and it prefigures the whole state-space and ARMA apparatus. Yule's least-squares fit of $u_x$ on its lags is the direct ancestor of the Yule-Walker equations and of every AR/VAR estimator in the wiki; reading it clarifies why AR models "borrow" periodicity from complex characteristic roots rather than assuming it. For a wiki heavy on VARs, unit roots, and spurious regression, this is the historical root of the univariate autoregression those all extend, and the sunspot series it launched remains the canonical nonlinear/periodic time-series benchmark.