ON THE APPROXIMATION OF CONTINUOUS-TIME THRESHOLD ARMA PROCESSES

Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in sever al recent papers by Brockwell ef al. (1991, Statist. Sinica, 1, 401-41 0), Tong and Yeung (1991, Statist. Sinica, 1, 411-430), Brockwell and Hyndman (1992, International Journal Forecasting, 8, 157-173) and Broc kwell (1994, J. Statist. Plann. Inference, 39, 291-304). A threshold A RMA process with boundary width. 2 delta > 0 is easy to define in term s of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipsckitz. The positive bo undary-width is a convenient mathematical device to smooth out the coe fficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equa tion from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with delta = 0 in the import ant case when only the autoregressive coefficients change with the lev el pf the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributi ons of the process in terms of the weak solution of a certain stochast ic differential equation. It is shown that the joint distributions of this solution with delta = 0 are the weak limits as 6 1 0 of the distr ibutions of the solution with delta > 0. The sense in which the approx imating sequence of processes used by Brockwell and Hyndman (1992, Int ernational Journal Forecasting, 8, 157-173) converges to this weak sol ution is also investigated. Some numerical examples illustrate the val ue of the latter approximation in comparison with the more direct repr esentation of the process obtained from the Cameron-Martin-Girsanov fo rmula. It is used in particular to fit continuous-time threshold model s to the sunspot and Canadian lynx series.