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
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 ..
6). 5 recreates Yule’s Fig. 3, . . 4], and quite slightly to decrease the maximum correlation attainable; it is not until the sampleinterval becomes as large as half a period, or thereabouts, that the contours of the curve round off and the maximum undergoes a rather sudden drop. , page 8) Yule then used these curves to construct the frequency distribution of the correlation coefficient for a given value of 2h/n. 6 and led Yule to conclude that the3 answer to our question, how the distribution of isolated frequencies at +1 and −1 closes up to the distribution of an isolated clump of frequency at zero, is then that the distribution first of all becomes a U-shaped distribution, with limits not far from +1 and −1, and that these limits, at first gradually and then more rapidly, close in on zero; but the distribution always remains U-shaped, and values of the correlation as far as possible removed from the true value (zero) always remain the most frequent.
21 present such a smooth appearance?