Abstract
Many time series show directionality as plots against time and against time-to-go are qualitatively different. A stationary linear model with Gaussian noise is non-directional (reversible). Directionality can be emulated by introducing non-Gaussian errors or by using a nonlinear model. Established measures of directionality are reviewed and modified for time series that are symmetrical about the time axis. The sunspot time series is shown to be directional with relatively sharp increases. A threshold autoregressive model of order 2, TAR(2) is fitted to the sunspot series by (nonlinear) least squares and is shown to give an improved fit on autoregressive models. However, this model does not model closely the directionality, so a penalized least squares procedure was implemented. The penalty function included a squared difference of the discrepancy between observed and simulated directionality. The TAR(2) fitted by penalized least squares gave improved out-of-sample forecasts and more realistic simulations of extreme values.
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In the following, “differences” refers to these lag one differences.
References
Beare, B.K., Seo, J.: Time irreversible copula-based Markov models. Econ. Theory 30, 1–38 (2012)
Chatfield, C.: The Analysis of Time Series: An Introduction, 6th edn., pp. 218–219, 223–224, 44–45. Chapman and Hall/CRC, London/Boca Raton (2004)
Lawrance, A.: Directionality and reversibility in time series. Int. Stat. Rev./Revue Internationale de Statistique 59(1), 67–79 (1991)
Mansor, M.M., Green, D.A., Metcalfe, A.V.: Modelling and simulation of directional financial time series. In: Proceedings of the 21st International Congress on Modelling and Simulation (MODSIM 2015), pp. 1022–1028 (2015)
Mansor, M.M., Glonek, M.E., Green, D.A., Metcalfe, A.V.: Modelling directionality in stationary geophysical time series. In: Proceedings of the International Work-Conference on Time Series (ITISE 2015), pp. 755–766 (2015)
Nash, J.C.: On best practice optimization methods in R. J. Stat. Softw. 60(2), 1–14 (2014)
Solar Influences Data Analysis Center, Sunspot Index and Long-term Solar Observations. http://www.sidc.be/silso (last accessed 17 October 2015)
Soubeyrand, S., Morris, C.E., Bigg, E.K.: Analysis of fragmented time directionality in time series to elucidate feedbacks in climate data. Environ. Model Softw. 61, 78–86 (2014)
Wild, P., Foster, J., Hinich, M.: Testing for non-linear and time irreversible probabilistic structure in high frequency financial time series data. J. R. Stat. Soc. A. Stat. Soc. 177(3), 643–659 (2014)
Acknowledgements
We thank the School of Mathematical Sciences at the University of Adelaide for sponsoring the presentation of this work by Maha Mansor at ITISE 2015 in Granada. We would also like to thank the Majlis Amanah Rakyat (MARA), a Malaysian government agency for providing education sponsorship to Maha Mansor at the University of Adelaide, and the SIDC, World Data Center, Belgium, for data.
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Mansor, M.M., Glonek, M.E., Green, D.A., Metcalfe, A.V. (2016). Threshold Autoregressive Models for Directional Time Series. In: Rojas, I., Pomares, H. (eds) Time Series Analysis and Forecasting. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-28725-6_2
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DOI: https://doi.org/10.1007/978-3-319-28725-6_2
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