Notes 2 Yule: The Time–Correlation Problem, Nonsense Correlations, Periodicity and Autoregressions 1. William S. Gosset (1876–1937) was one of the most influential statisticians of the twentieth century. As an employee of Guinness, the famous Irish brewer of stout, he was precluded from publishing under his own name and thus took the pseudonym ‘Student’. Gosset worked primarily on experimental design and small-sample problems and was the inventor of the eponymous Student’s t-distribution. For further biographical details see Pearson (1950, 1990). 2. Yule admitted that he could not produce a proof of this result. It is, in fact, a special case of a more general result proved by Egon Pearson for a non-random series: see Pearson (1922, pages 37–40, and in particular his equation (xviii)). 3. The method used by Yule to produce his Figs 5–9 is discussed in Yule (1926, page 55). We approximate it here to recreate these distributions in our composite Figure 2.6. 4. The calculations required to construct Figure 2.7 are outlined in Yule (1926, page 56). 5. For a derivation of this result in the more general context of calculating ‘intraclass’ correlation, see Yule and Kendall (1950, §§11.40–11.41). 6. A small Gauss program was written to simulate Yule’s sampling procedure and to compute the results shown in Table 2.1 and later in Tables 2.4 and 2.7. Necessarily, the results differ numerically from those of Yule because of the sampling process. 7. Yule states that the maximum negative correlation is that between ‘terms 2 and 8 or 3 and 9, and is −0.988’, which is clearly a misreading of his Table VI, which gives correlations to three decimal places, unlike our Table 2.6, which retains just two to maintain consistency with earlier tables. 8. The index numbers themselves are used, rather than the smoothed Index of Fluctuations, which are often analyzed (see Mills, 2011a, §§3.8–3.9). The serial correlations were obtained using the correlogram view in EViews 6. Compared to Yule’s own heroic calculations, which he described in detail (Yule, 1926, page 42) and reported as Table XIII and Fig. 19, they are uniformally smaller but display an identical pattern. 9. Yule (1927, pages 284–6) discussed further features of the sunspot numbers and their related disturbances estimated from equation (2.19). These features do not seem to appear in the extended series and are therefore not discussed here. 10. Indeed, the unusual behavior of sunspots over the last decades of the twen- tieth century has been the subject of great interest: see, for example, Solanki et al. (2004). 397 398 Notes 3 Kendall: Generalizations and Extensions of Stationary Autoregressive Models 1. Spencer-Smith (1947, page 105) returned to this point about moving averaging inducing spurious oscillations, but defined trend in a manner rather different to that considered here, being more akin to a long cycle: ‘such series do not contain very prolonged steady increases or decreases in the general values of these terms, as may happen in economic series, and where such movements occur the use of the moving average method may be valid’. 2. The use of the term autocorrelation follows Wold, 1938, with ‘serial correla- tion’ being reserved for the sample counterpart, rk. Note also the simplification of the notation used by Kendall from that employed in Chapter 2. 3. Hall (1925) had earlier suggested the same approach but restricted attention to local linear trends and failed to make the link with moving averages as set out below. Interestingly, however, he introduced moving sums to remove seasonal and cyclical components, referring to these as moving integrals and the process of computing them as moving integration, thus predating the use of the term integrated process to refer to a cumulated variable, introduced by Box and Jenkins (see §6.9), by some four decades. 4. A recent application of the approach is to be found in Mills (2007), where it is used to obtain recent trends in temperatures. An extension was suggested by Quenouille (1949), who dispensed with the requirement that the local poly- nomial trends should smoothly ‘fit together’, thus allowing discontinuities in their first derivatives. As with many of Quenouille’s contributions, this approach was both technically and computationally demanding and does not seem to have captured the attention of practitioners of trend fitting! 5. The data are provided in Table 1 of Coen et al. (1969). We have used EViews 6 to compute this regression, which leads to some very minor differences in the estimates as reported in Coen et al.’s Table 2 equation (7). 6. The estimates differ somewhat from Coen et al. (1969, Table 3, equation (10)) as some of the early observations on the lagged regressors are not provided there, so that the regression is here estimated over a slightly truncated sample period. 4 Durbin: Inference, Estimation, Seasonal Adjustment and Structural Modelling 1. Walker (1961, Appendix) provided a more complicated adjustment for the bias in this estimator. For this simulation it leads to a bias adjustment of the order 0.02, which would almost eradicate the bias. He also showed that Durbin’s adjustment would be approximately 0.004, which is what we find. 2. Durbin recounts that the government of the time was becoming increasingly worried about the rising unemployment figures. The Prime Minister, Harold Wilson, ‘had worked as an economic statistician in the government service during the war, and he was really rather good at interpretation of numerical data … and … was very interested in looking at the figures himself. He got the idea … that maybe the reason why the unemployment series appeared to Notes 399 be behaving in a somewhat strange way was due to the seasonal adjustment procedure that was being used … and asked the CSO to look into this. … It turned out that Wilson was right and there was something wrong with the seasonal adjustment. I think it is remarkable that a point like this should be spotted by a prime minister’ (Phillips, 1988, pages 140–1). To be fair, Wilson had a long-standing interest in economics and statistics, having been a lecturer in economic history at New College, Oxford (from the age of 21) and a Research Fellow at University College, as well as holding a variety of government posts that dealt with economic data. 3. The paper represented the work of a team of statisticians at the Research Branch of the CSO, with the authors being responsible for the supervision of the work and the writing of the paper. The Bureau of the Census X-11 seasonal adjust- ment package has been used extensively by statistical agencies across the world for adjusting economic time series. Mills (2011a, §§14.7–14.10) discusses its development in some detail. The Henderson moving average is based on the requirement that it should follow a cubic polynomial trend without distor- tion. The filter weights are conveniently derived in Kenny and Durbin (1982, Appendix) and also in Mills (2011a, §10.6). 4. Extensions to autoregressive models were later provided by Kulperger (1985) and Horváth (1993) and then to ARMA processes by Bai (1994). 5. Gauss’ original derivation of the recursive least squares (RLS) algorithm is given in Young (1984, Appendix 2), which provides the authorized French trans- lation of 1855 by Bertrand along with comments by the author designed to ‘translate’ the derivation into modern statistical notation and terminology. Young (2011, Appendix A) provides a simple vector-matrix derivation of the RLS algorithm. 6. Hald (1981) and Lauritzen (1981, 2002) claim that T.N. Thiele, a Danish astronomer and actuary, proposed in an 1880 paper a recursive procedure for estimating the parameters of a model that contained, as we know them today, a regression component, a Brownian motion and a white noise, that was a direct antecedent of the Kalman filter. Durbin was aware of this histori- cal link, for he remarked that it was ‘of great interest to note that the Kalman approach to time series modelling was anticipated in a remarkable way for a particular problem by … Theile in 1880, as has been pointed out by Lauritzen (1981)’ (Durbin, 1984, page 170). 7. The wearing of seatbelts by front seat occupants of cars and light goods vehicles had been made compulsory in the UK on 31 January 1983 for an experimen- tal period of three years with the intention that Parliament would consider extending the legislation before the expiry of this period. The Department of Transport undertook to monitor the effect of the law on road casualties and, as part of this monitoring exercise, invited Durbin and Harvey to conduct an independent technical assessment of the statistical evidence. 8. This ‘opportunity for public discussion’ was facilitated by reading the paper at a meeting of the RSS. The resulting discussion, along with the author’s rejoinder, was then published along with the paper. While we do not refer to the discussion here, it does make for highly entertaining and, in a couple of places, rather astonishing, reading, with non-statisticians being somewhat perplexed with the findings, to say the least! More generally, a number of Durbin’s papers published in RSS journals are accompanied with discussants 400 Notes remarks and author rejoinders and these typically make fascinating, if often tangential, reading. 9. Only Durbin and Koopman’s classical solution to the estimation problem is discussed here, although they also provide formulae for undertaking Bayesian inference (see Durbin and Koopman, 2000). 5 Jenkins: Inference in Autoregressive Models and the Development of Spectral Analysis 1. The name white noise was coined by physicists and engineers because of its resemblance to the optical spectrum of white light, which consists of very narrow lines close together.