Likelihood theory for the Graph Ornstein-Uhlenbeck process
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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 process to study the momentum and network effects amongst nodes. We distinguish the cases of the network-level GrOU and the node-level GrOU processes where the latter allows for the directed graph edges to be node-dependent. Given general likelihood frameworks, we derive maximum likelihood estimators and their usual properties (existence, uniqueness, consistency and efficiency). To quantify the estimation uncertainty, we present two novel central limit theorems under general assumptions with closed-form covariance matrices as the time horizon goes to infinity. Finally, we extend the -driven case to include a stochastic volatility modulation term and show that the central limit theorems still hold.
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