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 addresses a fundamental limitation of standard conditional heteroscedasticity models such as ARCH and GARCH: they assume that the model parameters are constant over the entire sample period. In practice, financial time series are subject to structural breaks, regime shifts, and gradual parameter drift caused by policy changes, crises, or market evolution. To cope with this, the authors propose an adaptive pointwise estimation framework that relaxes the global homogeneity assumption and instead seeks, for each observation time, the longest recent interval over which the data can be regarded as generated by a fixed‑parameter parametric model.

The methodology proceeds as follows. For a given time point (T) a candidate historical interval (I(T)=