Trimbur and William R. McElroy and Scott H. Hotta and Ruey S. Holan and Tucker S. Tu, Dong Song, F.

### General Introduction

Jay Breidt, Theodore W. William R. Bell, Ph. Census Bureau. He is a recognized researcher in the area of modeling and adjustment of seasonal economic time series. Scott H.

## Economic Time Series: Modeling and Seasonality

Holan, Ph. He is the author of over 30 articles on topics of time series, spatio-temporal methodology, Bayesian methods and hierarchical models. His work is largely motivated by problems in federal statistics, econometrics, ecology and environmental science. Tucker S. McElroy, Ph. His research is focused primarily upon developing novel methodology for time series problems, such as model selection and signal extraction. He has contributed to the model diagnostic and seasonal adjustment routines in the XARIMA seasonal adjustment program, and has taught seasonal adjustment to both domestic and international students.

In particular, some of the discussions about the interactions of time series modeling and seasonal adjustments are very enlightening and useful. We provide complimentary e-inspection copies of primary textbooks to instructors considering our books for course adoption. Most VitalSource eBooks are available in a reflowable EPUB format which allows you to resize text to suit you and enables other accessibility features. Where the content of the eBook requires a specific layout, or contains maths or other special characters, the eBook will be available in PDF PBK format, which cannot be reflowed.

For both formats the functionality available will depend on how you access the ebook via Bookshelf Online in your browser or via the Bookshelf app on your PC or mobile device. Stay on CRCPress. Exclusive web offer for individuals on all book. Preview this Book. Bell, Scott H.

Holan, Tucker S. However, whereas the t-distribution has fat-tails, and hence subjects extreme observations to a form of soft trimming, the EGB2 distribution has light tails but excess kurtosis and hence gives a gentle form of Winsorizing. The examples show that the EGB2 distribution can give a better fit to some macroeconomic series. The way in which DCS models respond to breaks was examined and it was shown that, contrary to what might be expected, they adjust almost as rapidly as Gaussian models. A seasonal adjustment procedure may be carried out with DCS models that include both trend and seasonal components.

Having fitted the DCS model, the scores are used to adjust the data before smoothing using a standard Gaussian model. Two or three iterations seem to be sufficient. The method was illustrated with data on tourists in Spain. There is a case for a structural break in April, , but the DCS model quickly adjusts to it and indeed it could be reasonably argued that it is better to let the change take place over several months rather than assigning it to just one.

In both cases a robust estimate of scale needs to be pre-computed and the process of computing M-estimates is then often iterated to convergence. Second, there is not an asymptotic theory for these estimators, and therefore inference procedures like tests or confidence regions are not available. We are grateful to the Gabriele Fiorentini and a referee for helpful comments on the first draft.

Skip to main content Skip to sections. Advertisement Hide. Download PDF. Robust time series models with trend and seasonal components. Open Access. First Online: 10 December The UC model in 1 is effectively in state space form and, as such, it may be handled by the Kalman filter KF ; see Harvey The time-varying parameter is then updated by a suitably defined filter. Such a model is said to be observation driven.

The above criteria are related to the behavior of the conditional score and whether or not it discounts large observations. This, in turn, connects to robustness, as shown in the sub-section following.

## How To Identify Patterns in Time Series Data: Time Series Analysis

The logarithm of a variable with a fat-tailed distribution has exponential tails. The exponential generalized beta distribution of the second kind EGB2 is obtained by taking the logarithm of a variable with a generalized beta distribution of the second kind GB2 distribution. Open image in new window. When first-order Gaussian models are fitted, there is little indication of residual serial correlation. There is excess kurtosis in all cases, but no evidence of asymmetry. Comparing the residuals with a fitted normal shows them to have a higher peak at the mean, as well as heavier tails; see Fig.

However, the differences between the two are small compared with the Gaussian model. Table 2 US Industrial production monthly. Table 3 US macroeconomic seriesâ€”Model comparison. In order to investigate the issue of adapting to a permanent change in level, an upward shift was added to the US industrial production data at the beginning of The size of the shift was calibrated so as to be proportional to the sample standard deviation of the series.

The results are shown in the Figs. For a one standard deviation shift the paths of all three filters after the break are similar. For two standard deviations the Gaussian filter adapts more quickly, but it is still well below the new level initially and after 9 or 10 months it is virtually indistinguishable from the other two filters.

### Working Papers & Publications

The simplest nonstationary unobserved components model is the Gaussian local level model, that is a random walk plus noise. In this case the Kalman filter is an exponentially weighted moving average EWMA of past observations. Stochastic trend and seasonal components may be introduced into Gaussian unobserved components models for location.

The BSM is made up of mutually independent trend, seasonal and irregular components. The pseudo-observations are now used to estimate the parameters in a BSM and the signal is estimated by smoothing. The automatic outlier detection option in STAMP finds five outliers, three of which are in , along with a small structural break which, although not sudden, is attributed to April The corresponding edited output is shown below. Fitting a BSM with dummy variables included for the outliers and break gives the smoothed components shown in Fig.

The filtered DCS-t trend shown in Fig. The figure also shows the seasonal component and contrasts the prediction errors irregular component and scores; it is evident that extreme observations are downweighted by the score. The autocorrelation functions of irregular and score are shown in Fig.

## Economic Time Series | Modeling and Seasonality | Taylor & Francis Group

The autocorrelations of the scores tend to be slightly bigger in absolute value, reflecting the fact that they are not weakened by outliers in the same way as the raw predictions errors. The trend at the second iteration is almost indistinguishable from the one obtained at the third. The result is shown in Fig.

Generally the results were similar except that the seasonal pattern for EGB2 was found to be deterministic. Acknowledgments We are grateful to the Gabriele Fiorentini and a referee for helpful comments on the first draft. Harvey AC Forecasting, structural time series models and the kalman filter. Harvey AC Dynamic models for volatility and heavy tails. Maravall A On structural time series models and the characterization of components.

Zhu D, Galbraith JW Modelling and forecasting expected shortfall with the generalised asymmetric student-t and asymmetric exponential power distributions. Bank of Italy Rome Italy 2. University of Cambridge Cambridge UK 3.

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- Economic Time Series: Modeling and Seasonality - CRC Press Book.
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