Hyper-Spherical and Elliptical Stochastic Cycles

13 Pages Posted: 19 Apr 2010

See all articles by Alessandra Luati

Alessandra Luati

University of Bologna - Department of Statistics

Tommaso Proietti

University of Rome II - Department of Economics and Finance

Date Written: 2010-02-27

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 rotation along a circle in the plane, and the reduced form is ARMA(2,1). 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 along an hyper-sphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of n-dimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, nā€‰āˆ’ā€‰1). The properties of the resulting models are analysed 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 time series literature.

Suggested Citation

Luati, Alessandra and Proietti, Tommaso, Hyper-Spherical and Elliptical Stochastic Cycles (2010-02-27). Journal of Time Series Analysis, Vol. 31, Issue 3, pp. 169-181, May 2010. Available at SSRN: https://ssrn.com/abstract=1591210 or http://dx.doi.org/10.1111/j.1467-9892.2010.00655.x

Alessandra Luati (Contact Author)

University of Bologna - Department of Statistics ( email )

Bologna, 40126
Italy

Tommaso Proietti

University of Rome II - Department of Economics and Finance ( email )

Via Columbia, 2
Rome, 00133
Italy

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