The Annals of Statistics

Yule’s “nonsense correlation” solved!

Philip A. Ernst, Larry A. Shepp, and Abraham J. Wyner

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In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically prove Yule’s 1926 empirical finding of “nonsense correlation” [J. Roy. Statist. Soc. 89 (1926) 1–63], which we do by analytically determining the second moment of the empirical correlation coefficient \begin{eqnarray*}&&\theta:=\frac{\int_{0}^{1}W_{1}(t)W_{2}(t)\,dt-\int_{0}^{1}W_{1}(t)\,dt\int_{0}^{1}W_{2}(t)\,dt}{\sqrt{\int_{0}^{1}W^{2}_{1}(t)\,dt-(\int_{0}^{1}W_{1}(t)\,dt)^{2}}\sqrt{\int_{0}^{1}W^{2}_{2}(t)\,dt-(\int_{0}^{1}W_{2}(t)\,dt)^{2}}},\end{eqnarray*} of two independent Wiener processes, $W_{1},W_{2}$. Using tools from Fredholm integral equation theory, we successfully calculate the second moment of $\theta$ to obtain a value for the standard deviation of $\theta$ of nearly 0.5. The “nonsense” correlation, which we call “volatile” correlation, is volatile in the sense that its distribution is heavily dispersed and is frequently large in absolute value. It is induced because each Wiener process is “self-correlated” in time. This is because a Wiener process is an integral of pure noise, and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of $\theta$, we offer implicit formulas for higher moments of $\theta$.

Article information

Ann. Statist. Volume 45, Number 4 (2017), 1789-1809.

Received: February 2016
Revised: August 2016
First available in Project Euclid: 28 June 2017

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Digital Object Identifier

Primary: 60J65: Brownian motion [See also 58J65] 60G15: Gaussian processes
Secondary: 60G05: Foundations of stochastic processes

Volatile correlation Wiener processes integral equations


Ernst, Philip A.; Shepp, Larry A.; Wyner, Abraham J. Yule’s “nonsense correlation” solved!. Ann. Statist. 45 (2017), no. 4, 1789--1809. doi:10.1214/16-AOS1509.

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