## The Annals of Statistics

### Yule’s “nonsense correlation” solved!

#### Abstract

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

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

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

https://projecteuclid.org/euclid.aos/1498636874

Digital Object Identifier
doi:10.1214/16-AOS1509

Zentralblatt MATH identifier
06773291

#### Citation

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. https://projecteuclid.org/euclid.aos/1498636874

#### References

• [1] Aldrich, J. (1995). Correlations genuine and spurious in Pearson and Yule.Statist. Sci.10364–376.
• [2] Boas, R. P. Jr. (1954).Entire Functions. Academic Press, New York.
• [3] Erdös, P. and Kac, M. (1946). On certain limit theorems of the theory of probability.Bull. Amer. Math. Soc.52292–302.
• [4] Feller, W. (1957).An Introduction to Probability Theory and Its Applications. Wiley, New York.
• [5] Granger, C. W. J. and Newbold, D. (1974). Spurious regression in econometrics.J. Econometrics2111–120.
• [6] Hendry, D. F. (1986). Economic modelling with cointegrated variables: An overview.Oxf. Bull. Econ. Stat.48201–212.
• [7] Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of self-normalized sums.Ann. Probab.1788–809.
• [8] Logan, B. F. and Shepp, L. A. (1968). Real zeros of random polynomials.Proc. Lond. Math. Soc.(3)1829–35.
• [9] Magnus, J. R. (1986). The exact moments of a ratio of quadratic forms in normal variables.Ann. Écon. Stat.495–109.
• [10] Mann, M. E., Zhang, Z., Hughes, M. K., Bradley, R. S., Miller, S. K., Rutherford, S. and Ni, F. (2008). Proxy-based reconstructions of hemispheric and global surface temperature variations over the past two millenia.Proc. Natl. Acad. Sci. USA10513252–13257.
• [11] McShane, B. B. and Wyner, A. J. (2011). A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?Ann. Appl. Stat.55–44.
• [12] Phillips, P. C. B. (1986). Understanding spurious regressions in econometrics.J. Econometrics33311–340.
• [13] Phillips, P. C. B. (1998). New tools for understanding spurious regressions.Econometrica661299–1325.
• [14] Rogers, L. C. G. and Shepp, L. A. (2006). The correlation of the maxima of correlated Brownian motions.J. Appl. Probab.43880–883.
• [15] Shepp, L. A. (1979). The joint density of the maximum and its location for a Wiener process with drift.J. Appl. Probab.16423–427.
• [16] Yule, G. U. (1926). Why do we sometimes get nonsense correlations between time series? A study in sampling and the nature of time series.J. Roy. Statist. Soc.891–63.