Adaptive pointwise estimation in time-inhomogeneous conditional heteroscedasticity models

This paper offers a new method for estimation and forecasting of the volatility of financial time series when the stationarity assumption is violated. Our general local parametric approach particularly applies to general varying-coefficient parametric models, such as GARCH, whose coefficients may arbitrarily vary with time. Global parametric, smooth transition, and change-point models are special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right-end point by a local change-point analysis. We construct locally adaptive estimates that can perform this task and investigate them both from the theoretical point of view and by Monte Carlo simulations. In the particular case of GARCH estimation, the proposed method is applied to stock-index series and is shown to outperform the standard parametric GARCH model.

💡 Research Summary

The paper tackles a fundamental limitation of conventional conditional heteroscedasticity models such as GARCH: the assumption that model coefficients remain constant over the entire sample period. In practice, financial time series frequently experience structural breaks, regime shifts, or smooth transitions that render the stationarity assumption untenable. Ignoring such non‑stationarity leads to biased volatility forecasts and unreliable risk measures.

To address this, the authors propose a fully data‑driven, locally adaptive estimation framework. The key idea is to treat each observation time t as a right‑hand endpoint and to search backward for the longest past interval (