Econometrica: Jan, 1969, Volume 37, Issue 1
Stochastic Nonlinear Models
L. R. Klein, R. S. Preston
The variance of solutions to stochastic linear dynamic systems tends to a finite limit if the system is damped. Limit cycles are possible in linear systems, but the corresponding solutions would have increasing variance in the stochastic case. On the other hand, nonlinear systems with limit cycles may have solutions with bounded variance in the stochastic case. A numerical example illustrating the limit cycles of Kaldor's nonlinear trade cycle model is shown to have solutions with bounded variance when the system is subjected to random shocks. The variance series is periodic with one-half the cyclical duration of the mean series. The amplitude of the mean series decreases over time. In a linear system, the mean series would have the same cyclical pattern as the nonstochastic solution.