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Continuous time threshold autoregressive models. (English) Zbl 0824.62082

Summary: The importance of nonlinear models in time series analysis has been recognized increasingly over the past ten years. A number of discrete- time nonlinear processes have been introduced and found valuable for the modelling of observed series. Among these processes are the discrete-time threshold models. The purpose of this paper is to define a continuous- time analogue of the threshold \(\text{AR} (p)\) process and to discuss some of its properties. For the continuous-time threshold \(\text{AR} (1)\) process we derive the stationary distribution (under appropriate assumptions) and consider problems of prediction and inference. The techniques developed apply equally well both to regularly and to irregularly spaced observations.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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