On The Approximation Of Moments For Continuous Time Threshold Arma Processes

. An approximating sequence of Markov processes with transitions at times 0, 1/n, 2/n,…, where n is large, was used in Brockwell and Hyndman (On continuous time threshold autoregression. Int. J. Forecasting 8 (1992), 157–73) and Brockwell (On continuous time threshold ARMA processes. J. Stat. Planning Inference 39 (1994). 291–304) to fit continuous time threshold autoregressive moving‐average (CTARMA) models with boundary width 2δ > 0 to both simulated and real data. In this paper we approximate CTARMA processes with δ= 0 by a new sequence of continuous processes and show that the distribution and conditional moments of these approximating processes converge to those of the process itself. This result provides us with a new method for estimating the conditional moments, which enables inference in such models. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the proces obtained from the Cameron‐Martin‐Girsanov formula (see, for example, Brockwell (On continuous time threshold ARMA processes. J. Stat. Planning Inference 39 (1994). 291–304) and Brockwell and Stramer (On the approximation of continuous time threshold ARMA processes. Ann. Inst. Statist. Math., to appear (1995))).