Time series and their methods of analysis are important subjects in statistics. Most of time series have a linear behavior and can be modelled by linear ARIMA models. However, some of realized time series have a nonlinear behavior and for modelling them one needs nonlinear models. For this, many good parametric nonlinear models such as bilinear model, exponential autoregressive model, threshold autoregressive model and GARCH model are commonly used.
On analysis of nonlinear time series, when there is no priori information, identification of a parametric nonlinear model, due to expanded nonlinear relation, is very difficult. A suitable substitution for parametric nonlinear models is the use of nonparametric models and their methods of analysis. In this direction various models and methods of analysis are introduced. One of the most important of such models is the functional-coefficient autoregressive (FAR) model. Flexibility of FAR models on fitting to real observation makes this model very useful as an applied methods in modelling economic survey, hydrology, and other related subjects. Many parametric models such as autoregressive model, threshold autoregressive model and exponential autoregressive model may be obtained as special cases of FAR models.
In this paper, using Chen and Tsay (1993) and Cai et al. (2000), we introduce FAR model and a method for fitting this model. Also, some methods for prediction of future of the time series using the fitted model are presented. Specially, a bootstrap predictive method for prediction of m-steps ahead of the time series is introduced. Using the bootstrap method point, interval and distribution of predictions are obtained.
We use FAR model for modelling of Iranian heavy crude oil price from July 1994 to December 2007. For this aim, using average prediction error criteria we identify a FAR (2,1) model with smoothing parameter h=1.3. This model is fitted using locally linear regression with Epanechnikov kernel. Examination of ability of prediction of the model and performance of the bootstrap prediction show that the fitted model has good performance. We finally use fitted model and bootstrap method for prediction of the first 3 months of 2008 of Iranian heavy crude oil price. Comparesion of these predictions with their real values shows a maximum of absolute differences of $4.75. This was related to March 2008 and it had a relative error of %5 with respect to the real value.