A proposed mildly explosive/self-exciting threshold autoregressive model applied to climatic time series

When selecting a time series model for a particular application it is appropriate to consider properties of the series under consideration. For example, it is not appropriate to apply standard autoregressive models to time series showing irreversibility, a feature observed in environmental series showing feedbacks. Models considered more suitable for modelling irreversible time series include Self-Exciting Threshold AutoRegressive (SETAR) models; a class of model composed of two or more regimes where the prediction equation used at a particular time is determined by some combination of previous time series values. In certain applications SETAR models are not able to reproduce rapid changes observed in series, as observed in the Burgundy Spring-Summer temperature anomaly series reconstructed from grape harvest dates [Chuine et al., Nature 432 (2004)]. To address this limitation, SETAR modified to have a regime capable of unstable behaviour flanked by two stable regimes are considered. One potentially suitable model class for the unstable middle regime is a mildly explosive model (MEM), an AR(1) model with a parameter greater than one which decays monotonically over time permitting unstable behaviour. MEM have found use in modelling financial exuberance and informed “collapsing bubble models” used to model rapid changes in the NASDAQ index. The SETAR class modified to have a MEM middle regime is termed here a MEMTAR. As an aid to evaluating the properties of a series, either historical or model derived, some descriptors of time series behaviour are proposed. These descriptors aim to detect a lack of directional symmetry in a series and inform the choice of model for a given application. A simulation study is used to obtain descriptors of values from a SETAR and a MEMTAR which each had an unstable order one autoregressive middle regime; a random walk for the SETAR and a less explosive version of a MEM for the MEMTAR. The simulation exercise results show that the MEMTAR tended to take extreme values more readily than the SETAR, suggesting the MEMTAR is worth further consideration. SETAR and MEMTAR models composed of autoregressive processes of orders one to three are applied to annual sunspot data from 1700 to 1988 and the reconstructed Burgundy temperature series of Chuine et al. from 1370 to 1977. Results are compared with those obtained from fitting standard autoregressive models. Results show that SETAR and MEMTAR models improve upon the ability of autoregressive models to model peak values.