Another note on the use of logarithmic time trend

REVISTA BRASILEIRA DE BIOMETRIA, 2021

The time series methodology is an important tool when using data over time. The time series can be composed of the components trend (Tt), seasonality (St) and the random error (at). The aim of this study was to evaluate the tests used to analyze the trend component, which were: Pettitt, Run, Mann-Kendall, Cox-Stuart and the unit root tests (Dickey-Fuller, Dickey-Fuller Augmented and Zivot and Andrews), given that there is a discrepancy between the test results found in the literature. The four series analyzed were the maximum temperature in the Lavras city, MG, Brazil, the unemployment rate in the Metropolitan Region of S~ao Paulo (RMSP), the Broad Consumer Price Index (IPCA) and the nominal Gross Domestic Product (GDP) of Brazil. It was found that the unit root tests showed similar results in relation to the presence of the stochastic trend for all series. Furthermore, the turning point of the Pettitt test diverged from all the structural breaks found through the Zivot and Andrews ...

J Forecasting, 1992

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 Q 1992 by John Wiley & Sons, Ltd.

Policymakers and investors often conceptualize trend growth as simply a medium/long term average growth rate. In practice, these averages are usually taken over arbitrary periods of time, thereby ignoring the large empirical growth literature which shows that doing so is inappropriate, especially in developing countries where growth is highly unstable. This paper builds on this literature to propose an algorithm, called “iterative Fit and Filter” (iFF), that extracts the trend as a sequence of medium/long term growth averages. iFF separates important country-specific historical episodes and trend growth durations - number of years between two consecutive trend growth shifts, vary substantially across countries and over time. We relate the conditional probabilities of up and down-shifts in trend growth next year to the country's current growth environment, level of development, demographics, institutions, economic management, and external shocks, and show how both iFF and the predictive model could be applied in practice.

2008

We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. 11 (2005), no. 15, 1305-1306.

Applied Stochastic Models in Business and Industry, 2009

We propose to decompose a financial time series into trend plus noise by means of the exponential smoothing filter. This filter produces statistically efficient estimates of the trend that can be calculated by a straightforward application of the Kalman filter. It can also be interpreted in the context of penalized least squares as a function of a smoothing constant has to be minimized by trading off fitness against smoothness of the trend. The smoothing constant is crucial to decide the degree of smoothness and the problem is how to choose it objectively. We suggest a procedure that allows the user to decide at the outset the desired percentage of smoothness and derive from it the corresponding value of that constant. A definition of smoothness is first proposed as well as an index of relative precision attributable to the smoothing element of the time series. The procedure is extended to series with different frequencies of observation, so that comparable trends can be obtained for say, daily, weekly or intraday observations of the same variable. The theoretical results are derived from an integrated moving average model of order (1, 1) underlying the statistical interpretation of the filter. Expressions of equivalent smoothing constants are derived for series generated by temporal aggregation or systematic sampling of another series. Hence, comparable trend estimates can be obtained for the same time series with different lengths, for different time series of the same length and for series with different frequencies of observation of the same variable. the analyst to study the behavior of each component separately, on the assumption that they have different causal forces.