🤖 AI Summary
Traditional GARCH models struggle to capture the long memory and state-dependent characteristics of financial volatility. This work proposes a novel state-dependent GARCH model that, for the first time, integrates a two-dimensional Markov chain with a latent-variable power-law kernel. By allowing the level and slope of the power-law kernel to evolve dynamically according to the underlying state, the model achieves adaptive decay of historical shocks. This approach effectively captures long-memory features through a low-dimensional state space and establishes rigorous ergodicity guarantees via Foster–Lyapunov theory. Simulation and empirical results demonstrate pronounced low-frequency persistence in log-squared innovations, and the model attains competitive out-of-sample forecasting accuracy using only a two-state specification.
📝 Abstract
This paper proposes a GARCH-type volatility model in which level-and-slope updates of a latent power-law kernel generate state-dependent decay of past shocks within a two-dimensional Markov state. We derive a joint Foster--Lyapunov condition and establish positive Harris recurrence and uniqueness of the invariant distribution. Simulations show substantial low-frequency persistence in log-squared innovations, especially near the diagnostic stability boundary. Empirically, the model captures a substantial portion of observed volatility persistence and delivers competitive out-of-sample forecast accuracy using only a two-dimensional Markov state.