By Terence C. Mills (auth.)

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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 ..

3. The correlations of terms adjacent to each other at either end of the sample are high and positive, but terms at opposite ends have moderately high and negative correlations. , page 18; italics in original). 4 Deviations from the mean of the sample in samples of 10 terms from a series with random differences, averaging separately samples in which (a) first deviation is +, (b) first deviation is −, (c) last deviation is +, (d) last deviation is −. The average of first or last deviations, respectively, called +1000 Term Expectation Experimental results a and b Term Experimental results c and d Together (1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 7 8 9 10 +1000 +684 +404 +158 −53 −228 −368 −474 −544 −579 +1000 +738 +436 +283 −30 −184 −346 −621 −655 −621 10 9 8 7 6 5 4 3 2 1 +1000 +754 +513 +274 +79 −194 −479 −498 −674 −776 +1000 +746 +474 +278 +25 −189 −412 −559 −664 −698 the appropriate correlation by the ratio of their standard deviations to the standard deviation of the first term.

8 display ‘no secular trend, and the whole movement is highly irregular. , page 26). , page 26). , page 26). 13 Having considered the various ‘internal’ properties of the different types of time series, Yule then turned his attention to his primary aim, that of analyzing the correlations between pairs of series drawn from each of the types. Using samples of size 10, he correlated 600 pairs of random series, 600 pairs of conjunct series with random differences, and 600 pairs of conjunct series with conjunct differences.