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

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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:  13 Jul 2016 18:32 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/15169 