In recent years there has been a continuing controversy over the relative merits to the Box-Jenkins or ARIMA method of time-series forecasting and the alternative state-space approach based on 'structural' or 'unobserved components' models. The paper by Garcia-Ferrer and del Hoyo (GH) makes an interesting contribution to this controversy, and it was felt that it could be used to highlight some of the important issues to readers of the Journal of Forecasting. Consequently, a distinguished actor in the controversy, Professor Andrew Harvey (H), whose computer program STAMP is used by GH in their paper, was asked to discuss the GH results. His comments are given below, together with a rejoinder by GH. I am sure that the debate could have been continued, with further comments from both sides, but I feel that the opposing views have been well represented and it is best to call a halt at this stage. However, the original paper and the additional comments make stimulating reading and may encourage other authors to contribute articles on the same subject. But the points at issue will not be easy to resolve in objective terms since, in a very real sense, the act of constructing the unobserved components model, within the state-space setting, introduces an inherent degree of subjectivity into model formulation. Moreover, the choice of the methodology is itself likely to be influenced by the assumptions made at this model-formulation stage.
Arguments about the correctness or otherwise of the detailed GH analysis apart, it seems that the major disagreements between GH and H hinge upon their different assumptions about the nature of the unobserved components and the models chosen to represent them. It is well known that the state-space description is not unique, depending, as it does, on the definition of the state variables chosen by the analyst (i.e. in the present setting, the assumed mathematical nature of the unobserved component models). Also, it is clear that the estimates of the components will be strongly dependent upon the assumptions about their dynamic nature, as specified by this a priori selected state-space model form. I t is here where GH and H make somewhat different, albeit individually quite reasonable, assumptions. Both discuss similar component models, but whereas GH contend that certain of the components should be considered as quasi-orthogonal, H does not see the need for this, and is happy if Maximum Likelihood estimation yields components which are quite highly correlated. Indeed, he seeks theoretical justification for such behaviour.
One problem is that GH are not very precise about their definition of 'orthogonality' and, in the main paper, they discuss it in relation to the significance, or otherwise, of the ccf between the residuals of the trend and cyclical components, respectively. H takes GH to task for this approach and shows, in his appendix, that one does indeed expect correlation between these estimates even if the disturbance terms are mutually uncorrelated. H is certainly correct, in this manner, to indicate the danger of an approach to orthogonality evaluation based on the statistical relationship between the residuals but, in doing so, he tends to divert attention away from the important conceptual point about orthogonality which GH are trying to make.
In their rejoinder, GH clarify their arguments by discussing the nature of the components and mentioning the idea of a 'smoothness prior'. They then point out that one way in which the smoothness of the trend estimate can be pre-specified is by modelling the trend as an Integrated Random Walk (IRW; see e.g. the discussion in Young and Ng, 1989;Ng and Young, 1990). Here, a white-noise input is only allowed to enter the second, slope equation of the model (i.e. u i = 0), and the associated Noise Variance Ratio (NVR; see also the above references) is suitably constrained to ensure a smoothly changing, lowfrequency trend estimate.
In contrast to this, we note that H allows random white-noise inputs to enter both the level and the slope equations of the IRW, which then constitutes the trend description in his Basic Structural Model