Transcription of YULE’S “NONSENSE CORRELATION” SOLVED!
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YULE’S “NONSENSE CORRELATION” SOLVED!
The Annals of Statistics2017, Vol. 45, No. 4, 1789 1809 Institute of Mathematical Statistics, 2017 yule S nonsense CORRELATION SOLVED! BYPHILIPA. ERNST ,LARRYA. SHEPP , 1 ANDABRAHAMJ. WYNER Rice University and University of Pennsylvania In this paper, we resolve a longstanding open statistical problem. Theproblem is to mathematically prove yule s 1926 empirical finding of non-sense correlation [J. Roy. Statist. (1926) 1 63], which we do byanalytically determining the second moment of the empirical correlation co-efficient := 10W1(t)W2(t) dt 10W1(t) dt 10W2(t) dt 10W21(t) dt ( 10W1(t) dt)2 10W22(t) dt ( 10W2(t) dt)2,of twoindependentWiener processes,W1,W2. Using tools from Fredholmintegral equation theory, we successfully calculate the second moment of toobtain a value for the standard deviation of of nearly The nonsense correlation, which we call volatile correlation, is volatile in the sense thatits distribution is heavily dispersed and is frequently large in absolute is induced because each Wiener process is self-correlated in time.
YULE’S “NONSENSE CORRELATION” SOLVED! 1793 2. A few results needed for obtaining the distribution of θ. In Section 2.1, we rewrite θ in an alternate form that will be useful in Section 3. The alternate form involves stochastic integrals rather than integrals of a Wiener process itself.
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