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ON THE APPROXIMATION OF CONTINUOUS TIME

THRESHOLD ARMA PROCESSES

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P. J. BROCKWELL AND O. STRAMER

*Statistics Department, Colorado State University, Fort
Collins,
CO 80521, U.S.A.*
(Received May 11, 1993; revised May 6, 1994)

**Abstract.**
Threshold autoregressive (AR) and autoregressive
moving average (ARMA) processes with continuous time parameter have
been discussed in several recent papers by Brockwell *et
al.*(1991, *Statist. Sinica*, **1**, 401-410), Tong
and Yeung (1991, *Statist. Sinica*, **1**, 411-430),
Brockwell and Hyndman (1992, *International Journal
Forecasting*, **8**, 157-173) and Brockwell (1994,
*J. Statist. Plann. Inference*, **39**, 291-304). A
threshold ARMA process with boundary width 2*delta* > 0 is easy to
define in terms of the unique strong solution of a stochastic
differential equation whose coefficients are piecewise linear and
Lipschitz. The positive boundary-width is a convenient mathematical
device to smooth out the coefficient changes at the boundary and
hence to ensure the existence and uniqueness of the strong solution
of the stochastic differential equation from which the process is
derived. In this paper we give a direct definition of a threshold
ARMA processes with *delta* = 0 in the important case when only the
autoregressive coefficients change with the level of the process.
(This of course includes all threshold AR processes with constant
scale parameter.) The idea is to express the distributions of the
process in terms of the weak solution of a certain stochastic
differential equation. It is shown that the joint distributions of
this solution with *delta* = 0 are the weak limits as
*delta* \downarrow 0 of the distributions of the solution with
*delta* > 0. The sense in which the approximating sequence of
processes used by Brockwell and Hyndman (1992, *International
Journal Forecasting*, **8**, 157-173) converges to this
weak solution is also investigated. Some numerical examples
illustrate the value of the latter approximation in comparison with
the more direct representation of the process obtained from the
Cameron-Martin-Girsanov formula. It is used in particular to fit
continuous-time threshold models to the sunspot and Canadian lynx
series.

*Key words and phrases*:
Non-linear time series,
continuous-time ARMA process, threshold model, stochastic
differential equation, approximation of diffusion processes.

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