  We consider the parametric estimation of the driving L\'evy process of a
multivariate continuous-time autoregressive moving average (MCARMA) process,
which is observed on the discrete time grid $(0,h,2h,...)$. Beginning with a
new state space representation, we develop a method to recover the driving
L\'evy process exactly from a continuous record of the observed MCARMA process.
We use tools from numerical analysis and the theory of infinitely divisible
distributions to extend this result to allow for the approximate recovery of
unit increments of the driving L\'evy process from discrete-time observations
of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen
dependent on $N$, the length of the observation horizon, such that $N h_N$
converges to zero as $N$ tends to infinity, then any suitable generalized
method of moments estimator based on this reconstructed sample of unit
increments has the same asymptotic distribution as the one based on the true
increments, and is, in particular, asymptotically normally distributed.
