Highest‐density forecast regions for nonlinear and non‐normal time series models

Abstract

Forecast regions are a common way to summarize forecast accuracy. They usually consist of an interval symmetric about the forecast mean. However, symmetric intervals may not be appropriate forecast regions when the forecast density is not symmetric and unimodal. With many modern time series models, such as those which are non-linear or have non-normal errors, the forecast densities are often asymmetric or multimodal. The problem of obtaining forecast regions in such cases is considered and it is proposed that highest-density forecast regions be used. A graphical method for presenting the results is discussed.KEY WORDS non-linear time series; non-normal time series; highest density regions; forecast intervals; threshold models Forecast regions are crucial to practical forecasting-they convey in a simple way the accuracy of a forecast. When the forecast densities are normal, as with ARIMA models (see, for example, Brockwell and Davis, 1991, p.175), they can easily be summarized by a forecast interval symmetric about the mean. For example, an approximate 95% forecast interval is the mean plus and minus two standard deviations. However, when a model is non-linear or nonnormal, the forecast densities are usually no longer normal and are frequently asymmetric or multimodal (Al-Qassam and Lane, 1989; Moeanaddin and Tong, 1990;Tong, 1990). In this case, forecast regions symmetric around the mean can be quite misleading. Instead, forecast regions need to be constructed so as to convey the shape of the forecast density. While there has been much activity in studying model properties, methods of estimation, diagnostics and point forecasts for many of the models that have been suggested (see, for example, De Gooijer and Kumar, 1992; Granger and Terasvirsta, 1993;Tong, 1990), forecast distributions and corresponding forecast regions tend to have been overlooked. In particular, the relative merits of different forecast regions do not seem to have been discussed previously in the forecasting literature. (It is not an issue for linear normal models because the forecast densities are normal and so all the obvious methods of constructing forecast regions yield identical results.) Often the methods that are appropriate for linear normal models are carried across to the non-linear or non-normal context. For example, Tong and Moeanaddin (1988) consider forecasts from a threshold autoregressive model but give forecast intervals based on a number of standard deviations about the mean. More frequently, the variance, MSE, or some other measure of accuracy is tabulated but no forecast intervals are given (e.g. Ray, 1988).

25 years of time series forecasting

“…A review of density forecasting is provided by Tay & Wallis (2000), along with several other articles in the same special issue of the JoF. Summarizing, a density forecast has been the subject of some interesting proposals including "fan charts" (Wallis, 1999) and "highest density regions" (Hyndman, 1995). The use of these graphical summaries has grown rapidly in recent years as density forecasts have become relatively widely used.…”

Recurrent Cartesian Genetic Programming of Artificial Neural Networks

“…The exponential smoothing used in this paper is from the Forecast package [27] for the R programming language [51]. When creating exponential smoothing models the ets function is used to find suitable parameters using the methods described in [25].…”

“…The ARIMA implementation used in this paper is from the Forecast package [27] for the R programming language [51]. When creating ARIMA models the auto.arima function [24] is used to find suitable p, d and q parameters as well as further sub parameters associated with the specific model.…”

25 Years of IIF Time Series Forecasting: A Selective Review

“…A review of density forecasting is provided by Tay & Wallis (2000), along with several other articles in the same special issue of the JoF. Summarizing, a density forecast has been the subject of some interesting proposals including "fan charts" (Wallis, 1999) and "highest density regions" (Hyndman, 1995).…”