Prediction intervals in linear and nonlinear time series with sieve bootstrap methodology

Forecasting is one of the main goals in time series analysis and it has had a great development in the last decades. In forecasting, the prediction intervals provide additional assessment of the uncertainty compared with a point forecast, which can better guide risk management decisions. The construction of prediction intervals requires fitting a model and the knowledge of the distribution of the observed data, which is typically unknown. Hence, data are usually assumed to follow some hypothetical distribution, and the resulting prediction interval can be adversely affected by departures from that assumption (Thombs and Schucany, J Am Stat Assoc 85:486–492, 1990). For this reason, in the last two decades several works based on free distributions have been proposed as an alternative for the construction of prediction intervals. Some alternatives consist in the sieve bootstrap approach, which assumes that the linear process admits typically an autoregressive AR representation, and it generates “new” realizations from the same model but with the resampled innovations (Alonso et al., J Stat Plan Inference 100:1–11, 2002; Chen et al., J Forecast 30:51–71, 2011). The linear nature of the models has not limited the implementation of the sieve bootstrap methodology in nonlinear models such as GARCH, since the squared returns can also be represented as linear ARMA process (Shumway and Stoffer, Time series analysis and its applications with R examples (2nd ed.). New York: Springer, 2006; Francq and Zakoian, GARCH Models: Structure, statistical inference and financial applications. Chichester: Wiley, 2010; Chen et al., J Forecast 30:51–71, 2011).