Séminaire Macroéconomie

Jeudi 08 octobre 2026

10h-12h

Salle 3001

Sebastian KUHNERT (Ruhr University Bochum) “New Developments in Nonlinear Time Series in Hilbert Spaces: Operator-Valued ARCH and Bilinear Processes”

Summary :

Linear processes in Hilbert spaces, particularly ARMA-type models, have been extensively studied in the literature. In contrast, nonlinear processes remain comparatively unexplored, despite a rich literature in univariate and multivariate settings. While functional (G)ARCH processes have received increasing attention in function spaces, bilinear processes in Hilbert spaces have so far only been touched upon, leaving their systematic development largely open. This talk presents recent developments for both model classes.

We introduce an operator-valued ARCH framework for general Hilbert spaces that, unlike existing functional ARCH and GARCH models, allows the full conditional covariance operator to evolve over time. For a simplified Constant Conditional Correlation specification, we establish stationarity, moment and weak dependence properties, and derive consistent Yule–Walker-type estimators. As work in progress, we further develop bilinear processes of general orders in Hilbert spaces, study their properties and stability conditions, and discuss their potential as a complementary or competing approach to ARCH-type models.