Yule’s “nonsense correlation”: Moments and density

Abstract

In 1926, G. Udny Yule (J. R. Stat. Soc. 89 (1926) 1–63) considered the following problem: given a sequence of pairs of random variables $\{X_k,Y_k\}$ ($k=1,2,\dots ,n$), and letting $X_i=S_i$ and $Y_i=S_i^{\prime }$ where $S_i$ and $S_i^{\prime }$ are the partial sums of two independent random walks, what is the distribution of the empirical correlation coefficient

$${\mathit{\rho }}_n=\frac{{\sum }_{i=1}^n{S}_i{S}_i^{\prime }-\frac{1}{n}({\sum }_{i=1}^n{S}_i)({\sum }_{i=1}^n{S}_i^{\prime })}{\sqrt{{\sum }_{i=1}^n{S}_i^2-\frac{1}{n}{({\sum }_{i=1}^n{S}_i)}^2}\sqrt{{\sum }_{i=1}^n{(S_i^{\prime })}^2-\frac{1}{n}{({\sum }_{i=1}^n{S}_i^{\prime })}^2}}?$$

Yule empirically observed the distribution of this statistic to be heavily dispersed and frequently large in absolute value, leading him to call it “nonsense correlation.” This unexpected finding led to his formulation of two concrete questions, each of which would remain open for more than ninety years: (i) Find (analytically) the variance of ${\mathit{\rho }}_n$ as $n\to \mathrm{\infty }$ and (ii): Find (analytically) the higher order moments and the density of ${\mathit{\rho }}_n$ as $n\to \mathrm{\infty }$. In 2017, Ernst, Shepp and Wyner (Ann. Statist. 45 (2017) 1789–1809) considered the empirical correlation coefficient

$$\mathit{\rho }:=\frac{{\int }_0^1{W}_1(t)W_2(t)dt-{\int }_0^1{W}_1(t)dt{\int }_0^1{W}_2(t)dt}{\sqrt{{\int }_0^1{W}_1^2(t)dt-{\left({\int }_0^1{W}_1(t)dt\right)}^2}\sqrt{{\int }_0^1{W}_2^2(t)dt-{\left({\int }_0^1{W}_2(t)dt\right)}^2}}$$

of two independent Wiener processes $W_1,W_2$, the limit to which ${\mathit{\rho }}_n$ converges weakly, as was first shown by P.C.B. Phillips (J. Econometrics 33 (1986) 311–340). Using tools from integral equation theory, Ernst, Shepp and Wyner (Ann. Statist. 45 (2017) 1789–1809) closed question (i) by explicitly calculating the second moment of ρ to be .240522. This paper adopts a completely different approach to the same question, rooted in an earlier literature on the laws of quadratic functionals of Gaussian diffusions (in particular, (Adv. in Appl. Probab. 25 (1993) 570–584; Stoch. Stoch. Rep. 41 (1992) 201–218)). This allows us to develop an Itô-formula approach from which we calculate expressions for the Laplace transform of ρ, leading to expressions for the moments which we evaluate up to order 16, thereby closing question (ii). This leads, for the first time, to an approximation to the density of Yule’s nonsense correlation. The broad applicability of this approach is demonstrated by answering the corresponding questions when the pair of independent Brownian motions is replaced by a pair of correlated Brownian motions, or by two independent Ornstein-Uhlenbeck processes, or by two independent Brownian bridges. We conclude by extending the definition of ρ to the time interval $[0,T]$ for any $T>0$ and prove a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes.