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The aim of this paper is to develop estimation and inference methods for thedrift parameters of multivariate L\'evy-driven continuous-time autoregressiveprocesses of order $p\in\mathbb{N}$. Starting from a continuous-timeobservation of the process, we develop consistent and asymptotically normalmaximum likelihood estimators. We then relax the unrealistic assumption ofcontinuous-time observation by considering natural discretizations based on acombination of Riemann-sum, finite difference, and thresholding approximations.The resulting estimators are also proven to be consistent and asymptoticallynormal under a general set of conditions, allowing for both finite and infinitejump activity in the driving L\'evy process. When discretizing the estimators,allowing for irregularly spaced observations is of great practical importance.In this respect, CAR($p$) models are not just relevant for "true"continuous-time processes: a CAR($p$) specification provides a naturalcontinuous-time interpolation for modeling irregularly spaced data - even ifthe observed process is inherently discrete. As a practically relevantapplication, we consider the setting where the multivariate observation isknown to possess a graphical structure. We refer to such a process as GrCAR anddiscuss the corresponding drift estimators and their properties. The finitesample behavior of all theoretical asymptotic results is empirically assessedby extensive simulation experiments.

Estimation and Inference for Multivariate Continuous-time Autoregressive Processes