By Terence C. Mills (auth.)
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Additional info for A Very British Affair: Six Britons and the Development of Time Series Analysis During the 20th Century
8 The correlations are all positive, as they evidently must be in a series that sweeps up from values round about 20 or 30 in earlier years to 100, 200 and over in the later years. 30] at r(40). , pages 42–3; notation altered for consistency) Yule’s next step was to compute the serial correlations of various differences of the index. 17 Serial correlations up to r(40) for Beveridge’s index numbers of wheat prices in Western Europe, 1545–1844 on noting that if k < h, r(k − h) = r(h − k). 18. Yule then embarked on a detailed discussion of the oscillations contained in the plots of these serial correlations, which we summarize thus.
Page 60; italics in original) Yule next considered the case when the differences are correlated such that 1 rx (k) is a linear function of k. This can be expressed as 1 rx (k) = 1 − αk since 1 rx (0) = 1. 9) j=1 To utilize this result, Yule constructed a series with correlated differences by taking the random series ut and cumulating 11-period Yule: Nonsense Correlations 37 moving sums, that is, by calculating t t st = uj , sj = ut + 2ut−1 + · · · +2ut−10 + ut−11 , xt = j=t−10 t = 11, . . , T j=1 It is then straightforward to show that 1 rx (k) Thus, setting α = = rs (k) = 1 11 1 − (k/11) for k = 1, .
XT is a zero mean series with standard deviation σx for which the serial correlations are rx (1), rx (2), . . 9. Then, if T is 32 A Very British Affair assumed to be large, T −1 T −1 (xt+1 − xt )2 = t=1 T −1 x2t+1 + t=1 T −1 x2t − 2 t=1 T −1 ≈2 T −1 x2t − 2 t=1 xt+1 xt t=1 T −1 =2 xt+1 xt t=1 T −1 t=1 xt+1 xt T −1 2 t=1 xt x2t 1 − t=1 or (cf. 7) Suppose that the differences are random, so that all the 1 rx (k) are zero and 2 rx (k + 1) = 0 for all k, implying that rx (k) = 2rx (k − 1) − rx (k − 2) Yule: Nonsense Correlations 33 Successive serial correlations are then generated by the arithmetical progression rx (2) = 2rx (1) − rx (0) = 2rx (1) − 1 rx (3) = 2rx (2) − rx (1) = 3rx (1) − 2 ..
A Very British Affair: Six Britons and the Development of Time Series Analysis During the 20th Century by Terence C. Mills (auth.)