The paper discusses a new, fully recursive approach to the adaptive modelling, forecasting and seasonal adjustment of nonstationary economic time-series. The procedure is based around a time variable parameter (TVP) version of the well known "component" or "structural" model. It employs a novel method of sequential spectral decomposition (SSD), based on recursive state-space smoothing, to decompose the series into a number of quasi-orthogonal components. This SSD procedure can be considered as

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... complete approach to the problem of model identification and estimation, or it can be used as a first step in maximum likelihood estimation. Finally, the paper illustrates the overall adaptive approach by considering a practical example of a U.K. unemployment series which exhibits marked nonstationarity caused by various economic factors. where t(k) is a low frequency or trend component; p(k) is a periodic or seasonal component; n(k) is a general stochastic perturbation component; and e(k) is a zero mean, serially uncorrelated white noise component, with variance az. The model (1) is appropriate for economic data exhibiting pronounced trend and periodicity and is the main vehicle utilised in the present paper for the development of adaptive seasonal adjustment procedures. The second model (2) can also be used to represent such heavily periodic time-series but it has much wider applicability to quasi-periodic and nonperiodic phenomena. It is utilised here mainly for the development of recursive state-space forcasting algorithms. Both models, however, are special cases of the general component model discussed in detail by Young (1988) and Ng and Young (1988) . Component models such as (1) and (2) have been popular in the literature on econometrics and forecasting (e.g. Nerlove et al., 1979; Bell and Hillmer, 1984) but it is only in the last few years that they have been utilised within the context of state-space estimation. Probably the first work of this kind was by Harrison and Stevens (1971, 1976) who exploited state-space methods by using a Bayesian interpretation applied to their "Dynamic Linear Model" (effectively a regression model with time variable parameters). More recent papers which exemplify this state-space approach and which are particularly pertinent to the present paper, are those of Jakeman and Young (1979, 1984), Kitagawa (1981) , Kitagawa and Gersch (1984), and Harvey (1984) . In the state-space approach, each of the components t(k), p(k) and n(k) is modelled in a manner which allows the observed time series y(k) to be represented in terms of a set of discrete-time state equations. And these state equations then form the basis for recursive state estimation, forecasting and smoothing. Before we investigate the use of these analytical techniques, therefore, it is appropriate to consider the specific form of the models for t(k), p(k) and n(k). where, 6k.j is the Kronecker delta function. Unless there is evidence to the contrary, Q, is assumed to be diagonal in form with unknown elements q, lJ and q,22, respectively. This GRW model subsumes, as special cases (see, for example, Young, 1984): the very well known and used random walk (RW: e = 1; B = 7 = 0; q,z(k) = 0); the smoothed random walk (SRW: fl = "l = l; 0 < ~ < 1.0; tltl(k ) = 0); and, most importantly in the present paper, the integrated