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Modelling trends and cycles in economic time series: historical perspective and future developments

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Abstract

This paper provides a retrospective on the modelling of trends and cycles in economic time series and considers where the research agenda currently stands and where future developments might lie. A brief survey of the early empirical research on trends and cycles is first provided before attention is focused on four papers published in 1961—our ‘annus mirabilis’ of trend and cycle modelling—which we argue have been ‘prime movers’ in various aspects of research in this area. The links from these papers to current research issues are then teased out before the likely future directions of research in both theoretical and applied aspects of the modelling of trends and cycles are considered.

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Notes

  1. In 1905, Einstein published four journal articles plus a doctoral dissertation on the photoelectric effect, Brownian motion and the special theory of relativity. These so ‘revolutionized the basic principles of physics’ and ‘had [such] far-reaching consequences for a physical understanding of the world’ (Renn and Hoffmann 2005, p S437) that this year was called Einstein’s ‘annus mirabilis’ (wonderful year): see also Stachel (1988). In fact, 1666 has a prior claim to such an accolade, being the year in which Newton observed the apple falling from the tree and thus discovered gravitation. The first known usage of the phrase is as the title of a poem by the English poet John Dryden about the events of 1666. Although this year was beset by great calamities (for example, plague, which led to Newton leaving Cambridge and retreating to his family home near Grantham, where he observed the eponymous falling apple, and the Great Fire of London), Dryden chose to interpret the absence of greater disasters as a miraculous intervention by God, since ‘666’ is the Number of the Beast and so the year 1666 was expected to be particularly disastrous. Much closer to 1961, Annus Mirabilis was also the title of a 1967 poem by Philip Larkin, which began with the famous lines ‘Sexual intercourse began/In nineteen sixty-three/(which was rather late for me/…’!.

  2. Usefully, the data used by Klein and Kosobud (annual U.S. series from 1900 to 1953) were tabulated in their paper, thus enabling modern day researchers to re-analyse the series using current techniques of time series econometrics that were unavailable at the time they undertook their research: see Mills (2008).

  3. The original and unpublished Office of Naval Research report outlining ‘Holt’s method’ of EWMA forecasting has recently been reprinted as Holt (2004a): retrospectives of this technique are provided by Holt (2002, 2004b).

  4. That the penalised least squares approach, in various forms, anteceded the HP filter by several decades was well known to Leser and to Hodrick and Prescott, although the latter appear to be unaware of Leser’s paper. Pedregal and Young (2001) provide both historical and multidisciplinary perspective.

  5. These filters should not be confused with the Henderson–Musgrave trend filters used in the seasonal adjustment algorithms of many central statistical agencies: see Kenny and Durbin (1982) and, for more recent developments, Gray and Thomson (2002) and Quenneville et al. (2003).

  6. Loosely speaking, Kalman (1960) analysed the case of discrete observations, while he and Bucy dealt with the technically more difficult continuous case.

  7. For an accessible and detailed exposition of Kalman filtering, and which also contains material on its prehistory, see Pollock (2003a), who claims that Kalman’s algorithm was, in fact, anteceded by Plackett (1950).

  8. The formulation of the trend in (11) is called the ‘contemporaneous state’ model by Harvey and Koopman (2000), in contrast to the ‘future state’ model, in which μ t+1 = μ t  + ξ t . When the innovations are assumed to be uncorrelated, as we do here, the two specifications are essentially identical, but interesting differences appear when the innovations are allowed to be correlated. Harvey and Koopman (2000) and Harvey and De Rossi (2006) provide detailed discussion of the correlated innovations case.

  9. This result follows from the application of the Weiner–Kolmogorov theory of signal extraction, the classic exposition of which for stationary series is Whittle (1983), with Bell (1984) extending the theory to nonstationary signals: see Pollock (2006) for a recent and extensive discussion.

  10. We ignore ‘end-point’ problems caused by having only a finite sample of observations available. Pollock (2000, 2001) proposes an ingenious method which is able to bypass such problems.

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    Terence C. Mills

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Mills, T.C. Modelling trends and cycles in economic time series: historical perspective and future developments. Cliometrica 3, 221–244 (2009). https://doi.org/10.1007/s11698-008-0031-y

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