Volume 1, Issue 3

This paper investigates the problem of testing for the symmetry of linear time series driven by asymmetric innovations. In particular, we examine the performance of alternative symmetry tests when innovations are fat tailed. Among the tests considered, only the test based on the tail estimator of the spectral measure yields satisfactory results in the presence of fat-tailed innovations.

The method of reconstructing an n-dimensional system from observations is to form vectors of m consecutive observations, which for m 2n, is generically an embedding. This is Takens's result. The Jacobian methods for Lyapunov exponents utilize a function of m variables to model the data, and the Jacobian matrix is constructed at each point in the orbit of the data. When embedding occurs at dimension m = n, the Lyapunov exponents of the reconstructed dynamics are the Lyapunov exponents of the original dynamics. However, if embedding only occurs for an m > n, then the Jacobian method yields m Lyapunov exponents, only n of which are the Lyapunov exponents of the original system. The problem is that as currently used, the Jacobian method is applied to the full m-dimensional space of the reconstruction, and not just to the n-dimensional manifold that is the image of the embedding map. Our examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system.

This paper tests for nonlinearity in EMS exchange rates using the bispectrum. The early experience of the ERM witnessed numerous realignments. We find that exchange rates follow a linear process over the period 1979-1987, consistent with the predictions of the realignment target zone model, where a stabilizing nonlinearity is absent. But from 1987-1992, no realignments occurred, and many currencies conformed to a nonlinear process, consistent with the credible target zone model where an inherent nonlinearity stabilizes exchange rates. However, the Italian lira and the Irish pound follow a linear process, which suggests that a target zone has not proven effective in stabilizing exchange rates.

This paper describes SIMANN, a Fortran and GAUSS implementation of the simulated annealing algorithm. The Fortran code was used in "Global Optimization of Statistical Functions with Simulated Annealing" (Goffe, Ferrier, and Rogers 1994). In that paper, simulated annealing was found to be competitive, if not superior, to multiple restarts of conventional optimization routines for difficult optimization problems. This paper compares SIMANN to the DFP algorithm on another optimization problem, namely, the maximum likelihood estimation of a rational expectations model, which was previously studied in the literature. SIMANN again performs quite well, and shows several advantages over DFP. This paper also describes simulated annealing, and gives explicit directions and an example for using the included GAUSS and Fortran code.