Likelihood theory for the Graph Ornstein-Uhlenbeck process

TL;DR

This paper develops a likelihood-based framework for modeling and inferring the structure of multivariate Graph Ornstein-Uhlenbeck processes driven by Lévy processes, including parameter estimation and graph structure recovery.

Contribution

It introduces maximum likelihood estimators for the GrOU process, proves their properties, and proposes a penalized likelihood method for joint parameter and graph structure inference.

Findings

MLEs are shown to be consistent and asymptotically normal.

The penalized likelihood approach accurately infers graph structure.

The theory extends to models with stochastic volatility.

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 L\'evy 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 L\'evy process is considered.