# Discipline of Operations Management and Econometrics

## Hyper-spherical and Elliptical Stochastic Cycles

Dr Tommaso Proietti, University of Tor Vegata, Italy

1st Nov 2010  10:00 am - Room 498, Merewether Building

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 2kΠ or (2k + 1) Π, 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 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, 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 mink-muskrat interaction. The hyper-spherical model provides an interesting re-interpretation of the cycle in US Gross Domestic Product quarterly growth and the cycle in the Fortaleza rainfall series.