Stability in Self-Exciting Threshold Autoregressive Models

Self-exciting threshold autoregressive (SETAR) models switch regimes according to their own past, so stability criteria based on unrestricted regime switching can be too conservative and sometimes misleading. This paper develops a stability diagnostic that respects the model’s endogenous switching rule. The key idea is to focus on the regime paths that are actually feasible under the SETAR dynamics, rather than all mathematically possible paths. We provide a computable search procedure that identifies the long-run switching patterns implied by the deterministic skeleton and uses them to assess whether the model is stable, explosive, or inconclusive. The analysis shows that local stability of each regime does not guarantee global stability, because feasible switching patterns can still generate explosive behavior; conversely, a locally unstable regime need not make the whole model unstable if the switching rule prevents explosive paths from occurring. Numerical examples and Monte Carlo evidence show that the method can deliver sharper stability conclusions than joint spectral radius criteria and is often decisive in empirically relevant cases.