Likelihood theory for the graph Ornstein-Uhlenbeck process

Valentin Courgeau () and Almut Veraart
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Valentin Courgeau: Imperial College London

Statistical Inference for Stochastic Processes, 2022, vol. 25, issue 2, No 2, 227-260

Abstract: Abstract We consider the problem of modelling restricted interactions between continuously-observed time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph Ornstein-Uhlenbeck (GrOU) process driven by a general Lévy process to study the momentum and network effects amongst nodes, effects that quantify the impact of a node on itself and that of its neighbours, respectively. We derive the maximum likelihood estimators (MLEs) and their usual properties (existence, uniqueness and efficiency) along with their asymptotic normality and consistency. Additionally, an Adaptive Lasso approach, or a penalised likelihood scheme, infers both the graph structure along with the GrOU parameters concurrently and is shown to satisfy similar properties. Finally, we show that the asymptotic theory extends to the case when stochastic volatility modulation of the driving Lévy process is considered.

Keywords: Ornstein-Uhlenbeck processes; Multivariate Lévy process; Continuous-time likelihood; Maximum likelihood estimator; Graphical modelling; Central limit theorem; Adaptive Lasso (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11203-021-09257-1

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