Luati, Alessandra and Proietti, Tommaso (2009): Hyperspherical and Elliptical Stochastic Cycles.

PDF
MPRA_paper_15169.pdf Download (290kB)  Preview 
Abstract
A univariate first order stochastic cycle can be represented as an element of a bivariate first order vector autoregressive process, or VAR(1), where the transition matrix is associated with a Givens rotation. From the geometrical viewpoint, the kernel of the cyclical dynamics is described by a clockwise rotation along a circle in the plane. The reduced form of the cycle is either ARMA(2,1), with complex roots, or AR(1), when the rotation angle equals 2k\pi or (2k + 1)\pi; k = 0; 1;... This paper generalizes this representation in two directions. According to the first, the cyclical dynamics originate from the motion of a point along an ellipse. The reduced form is also ARMA(2,1), but the model can account for certain types of asymmetries. The second deals with the multivariate case: the cyclical dynamics result from the projection along one of the coordinate axis of a point moving in Rn along an hypersphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of ndimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, n  1). The properties of the resulting models are analyzed in the frequency domain, and we show that this generalization can account for a multimodal spectral density. The illustrations show that the proposed generalizations can be fitted successfully to some well known case studies of the econometric and time series literature. For instance, the elliptical model provides a parsimonious but effective representation of the minkmuskrat interaction. The hyperspherical model provides an interesting reinterpretation of the cycle in US Gross Domestic Product quarterly growth and the cycle in the Fortaleza rainfall series.
Item Type:  MPRA Paper 

Original Title:  Hyperspherical and Elliptical Stochastic Cycles 
Language:  English 
Keywords:  State space models; PredatorPrey Interaction; Givens Rotations. 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models E  Macroeconomics and Monetary Economics > E3  Prices, Business Fluctuations, and Cycles > E32  Business Fluctuations ; Cycles C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 
Item ID:  15169 
Depositing User:  Tommaso Proietti 
Date Deposited:  12. May 2009 08:35 
Last Modified:  24. Aug 2015 07:56 
References:  Bulmer M.G. (1974), A statistical analysis of the 10year cycle in Canada, Journal of Animal Ecology, 43, 701718. Chan W.Y. T. and Wallis K.F. (1978), Multiple Time Series Modelling: Another Look at the MinkMuskrat Interaction, Applied Statistics, 27, 2, 168175. Doornik, J.A. (2006), Ox. An ObjectOriented Matrix Programming Language, Timberlake Consultants Press, London. Durbin, J., Koopman S.J. (2001), Time Series Analysis by State Space Methods, Oxford University Press. Givens W. (1958), Computation of plane unitary rotations transforming a general matrix to triangular form, SIAM Journal, 6, 1, 2650. Goldstein H. (1980), Classical Mechanics, second edition, AddisonWesley. Golub G.H. and van Loan C.F. (1996), Matrix Computations, third edition, The John Hopkins University Press. Gray, S., Zhang, N.F., Woodward, W.A., 1989. On generalized fractional processes. Journal of Time Series Analysis, 10, 233257. Hannan E.J. (1964), The Estimation of a Changing Sesasonal Pattern, Journal of the American Statistical Association, 59, 308, 10631077. Harvey A.C. (1985), Trends and Cycles in Macroeconomic Time Series, Journal of Business and Economics Statistics, 3, 3, 216227. Harvey A.C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press. Harvey A.C. and Jaeger A. (1993), Detrending, Stylized Facts and the Business Cycle, Journal of Applied Econometrics, 8, 231247. Harvey A.C. and Souza R.C. (1987), Assessing and Modeling the Cyclical Behavior of Rainfall in Northeast Brazil, Journal of Climate and Applied Meteorology, 26, 10, 13391344. Harvey A.C. and Trimbur, T.M. (2003), General Modelbased Filters for Extracting Cycles and Trends in Economic Time Series, The Review of Economics and Statistics, 85, 2, 244255. Haywood J. and TunnicliffeWilson, G. (2000), An Improved State Space Representation for Cyclical Time Series, Biometrika, 87, 3, 724726. Kendall M.G. (1945), On the Analysis of Oscillatory TimeSeries, Journal of the Royal Statistical Society, Series B, 108, 1/2, 93141. Koopman S.J., Harvey, A.C., Doornik, J.A. and Shephard, N. (2006), STAMP: Structural Time Series Analyser, Modeller and Predictor, Timberlake Consultants Press. Lancaster P. and Tismenetsky M. (1985), The Theory of Matrices, Academic Press. Luetkepohl H. (2006), New Introduction to Multiple Time Series Analysis, Springer. Meyer C.D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM. Morgan M.S. (1990), The History of Econometric Ideas. Cambridge University Press, Cambridge, U.K. Morley J.C., Nelson C.R., Zivot E. (2003), Why Are the BeveridgeNelson and Unobserved Components Decompositions of GDP So Different?, The Review of Economics and Statistics, 85, 2, 235243. Terasvirta, T. (1985), Mink and Muskrat interaction: a structural approach, Journal of Time Series Analysis, 6, 3, 171180. Tong H., Lim K.S. (1980), Threshold Autoregression, Limit Cycles and Cyclical Data, Journal of the Royal Statistical Society, Series B, 42, 3, 249292. Trimbur, T.M. (2005), Properties of Higher Order Stochastic Cycles, Journal of Time Series Analysis, 27, 1, 117. West, M., Harrison, J. (1989), Bayesian Forecasting and Dynamic Models, 1st edition, New York, SpringerVerlag. West, M., Harrison, J. (1997), Bayesian Forecasting and Dynamic Models, 2nd edition, New York, SpringerVerlag. Yule, G.U. (1927), On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolferâ€™s Sunspot Numbers, Philosophical Transactions of the Royal Society of London, Series A, Containing Papers of a Mathematical or Physical Character, 226, 1, 267298. Zellner A., Palm F. (1974), Time series analysis and simultaneous equation econometric models, Journal of Econometrics, 2, 1754. Zhang W., Yao Q., Tong H.,Stenseth N.C. (2003), Smoothing for Spatiotemporal Models and Its Application to Modeling MuskratMink Interaction, Biometrics, 59, 4, 813821. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/15169 