Sparse estimators for multivariate integer-valued autoregressive models with applications to inference for Hawkes processes

We investigate sparse and low-rank estimation methods for the intensity functions of multivariate Hawkes processes. While univariate Hawkes processes are approximated by integer-valued autoregressive models in the weak topology, we extend this approximation result to the multivariate setting. Building on this foundation, we develop new sparse and higher-order reduced-rank estimation procedures for multivariate Hawkes processes by leveraging methodology for multivariate integer-valued time series. In addition, we study a sparse weighted least squares estimator and establish error bounds for all proposed estimators. All theoretical results are derived by verifying novel moment and mixing conditions that ensure the applicability of concentration inequalities to weakly dependent data. Finally, we conduct an empirical study to assess the finite-sample performance of all estimation methods.