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The above model takes the threshold as given, and thus no attempt is made to estimate it. We will return to the estimation of thresholds later in the chapter. In addition to the threshold autoregressive models discussed so far, which are in discrete time, there are continuous time versions of these models. Often, continuous time modelling 24 Forecasting Volatility in the Financial Markets is easier for analysis and for obtaining analytical solutions. We will refer to some of the related literature without discussing these models: Tong and Yeung (1991), Brockwell and Hyndman (1992), Brockwell and Stramer (1992), Tong (1992) and Brockwell (1993).

In threshold models, the shifts between the regimes are modelled explicitly in terms of the time series under consideration. That is, if we have T , the switch between the regimes depends on either XT or a time series Xt t = 1 some other realization in the past, XT −d (or it could depend on another time series which is assumed to determine the shifts in the relevant time series). e. the introduction of regimes via a threshold. This idea is called the ‘threshold principle’, where a complex stochastic system is decomposed into simpler subsystems.

A1 and a2 are taken to be real constants, and it is assumed that t is a sequence of iid random variables with zero mean and constant variance 2 . This model could be referred to as an asymmetric AR model. Analogous asymmetric MA models are considered by Wecker (1981), and will be discussed later. Simulations of the model to determine the small-sample properties of the estimator yield the following results: 1. In general, aˆ 1 and aˆ 2 exhibit better overall performance when a1 and a2 are negative.

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