Dealing with Nonstationarity: Detrending, Smoothing and Differencing

Abstract

10.1 As we discussed in §§2.6–2.9, Hooker (1901b, 1905) was the first to be concerned with the problems of dealing with time series containing trends, proposing both differencing and the use of moving averages to ‘detrend’ the data prior to statistical analysis.¹ Beveridge (1921, 1922) later used a variation on the moving average to eliminate a secular trend from his wheat prices before subjecting them to periodogram analysis (§§3.8–3.9). The variate differencing approach examined in detail in Chapter 4 explored the link between successive differencing and fitting polynomials in time to a series, with Persons (1917) explicitly considering the decomposition of an observed time series into various unobserved components, one of which was the secular trend (§4.11). Indeed, the identification and removal of the trend component became a preoccupation of many analysts of time series data for much of the twentieth century, even though it was conceded that even the definition of a trend posed considerable conceptual problems: as Kendall (1941, page 43) remarked

(t)he concept of ‘trend’, like that of time itself, is one of those ideas which are generally understood but difficult to define with exactitude. A movement which has the evolutionary appearance of a trend over a period of thirty or forty years may in reality be one phase of an oscillatory movement of greater extent. A good deal depends on the length of the series under consideration whether we regard any particular tendency in the series as a trend, or a longterm movement, or an oscillation, or short-term movement. But in any case we require of a trend curve that it shall exhibit only the general direction of the time-series, and in practice this amounts to saying that it must be representable, at least locally, by a smooth non-periodic function such as a polynomial or a logistic curve.