  The aim of this paper is to develop estimation and inference methods for the
drift parameters of multivariate L\'evy-driven continuous-time autoregressive
processes of order $p\in\mathbb{N}$. Starting from a continuous-time
observation of the process, we develop consistent and asymptotically normal
maximum likelihood estimators. We then relax the unrealistic assumption of
continuous-time observation by considering natural discretizations based on a
combination of Riemann-sum, finite difference, and thresholding approximations.
The resulting estimators are also proven to be consistent and asymptotically
normal under a general set of conditions, allowing for both finite and infinite
jump 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 natural
continuous-time interpolation for modeling irregularly spaced data - even if
the observed process is inherently discrete. As a practically relevant
application, we consider the setting where the multivariate observation is
known to possess a graphical structure. We refer to such a process as GrCAR and
discuss the corresponding drift estimators and their properties. The finite
sample behavior of all theoretical asymptotic results is empirically assessed
by extensive simulation experiments.
