YULE'S NONSENSE CORRELATION SOLVED!

• 成果类型: Article
• 署名作者: Ernst, Philip A.; Shepp, Larry A.; Wyner, Abraham J.
• 署名单位: Rice University; University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1509
• 发表日期: 2017
• 页码: 1789-1809
• 关键词: understanding spurious regressions

摘要

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 theta : = integral(1)(0) W-1(t) W-2(t) dt - integral(1)(0) W-1(t) dt integral(1)(0) W-2(t) dt/root integral(1)(0) W-1(2)(t) dt - (integral(1)(0) W-1(t) dt)(2) root integral(1)(0) W-2(2)(t) dt - (integral(1)(0) W-2(t) dt)(2), 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.