Abstract
Consider two independent random walks. By chance, there will be spells of association between them where the two processes move in the same direction, or in opposite direction. We compute the probabilities of the length of the longest spell of such random association for a given sample size, and discuss measures like mean and mode of the exact distributions. We observe that long spells (relative to small sample sizes) of random association occur frequently, which explains why nonsense correlation between short independent random walks is the rule rather than the exception. The exact figures are compared with approximations. Our finite sample analysis as well as the approximations rely on two older results popularized by Révész (Stat Pap 31:95–101, 1990, Statistical Papers). Moreover, we consider spells of association between correlated random walks. Approximate probabilities are compared with finite sample Monte Carlo results.
Highlights
The puzzle why “we sometimes get nonsense-correlation between time-series” has first been addressed in the seminal paper by Yule (1926)
We observe that long spells of random association occur frequently, which explains why nonsense correlation between short independent random walks is the rule rather than the exception
In this note we focus on finite samples with a special interest on small sizes
Summary
The puzzle why “we sometimes get nonsense-correlation between time-series” has first been addressed in the seminal paper by Yule (1926). 33) provided experimental evidence, obtained by drawing playing cards from shuffled packs, that “The frequency-distribution of the correlations of samples of 10 observations [...] are much more widely dispersed than the correlations from samples of random series” His findings were accomplished by the computer experimental evidence on spurious regressions by Granger and Newbold (1974) for independent random walks of length 50, see Palm and Sneek (1984) for further Monte Carlo results. Yule (1926, Fig. 14) observed that random walks may trend in the same direction (concordance) or in the opposite direction (discordance) for certain periods of time This is an intuitive explanation for nonsense correlation: there will be cluster of association between independent random walks. The latter is evaluated numerically in Sect. Let logb stand for the logarithm to the base b, while ln denotes the natural logarithm
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