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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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4.4 Forecasting

Unlike the linear model, there exist no closed-form formulas to compute forecasts of most nonlinear models when the forecast horizon is greater than 1. We use parametric bootstraps to compute nonlinear forecasts. It is understood that the model used in forecasting has been rigorously checked and is judged to be adequate for the series under study. By a model, we mean the dynamic structure and innovational distributions. In some cases, we may treat the estimated parameters as given.

4.4.1 Parametric Bootstrap

Let T be the forecast origin and ℓ be the forecast horizon (ℓ > 0). That is, we are at time index T and interested in forecasting xT+ℓ. The parametric bootstrap considered computes realizations xT+1, … , XT+ℓ sequentially by (a) drawing a new innovation from the specified innovational distribution of the model, and (b) computing xT+i using the model, data, and previous forecasts xT+1, … , xT+i−1. This results in a realization for xT+ℓ. The procedure is repeated M times to obtain M realizations of xT+ℓ denoted by Inline. The point forecast of xT+ℓ is then the sample average of Inline. Let the forecast be xT(ℓ). We used M = 3000 in some applications and the results seem fine. The realizations can also be used to obtain an empirical distribution of xT+ℓ. We make use of this empirical ...

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