  In this paper we present a parametric estimation method for certain
multi-parameter heavy-tailed L\'evy-driven moving averages. The theory relies
on recent multivariate central limit theorems obtained in [3] via Malliavin
calculus on Poisson spaces. Our minimal contrast approach is related to the
papers [14, 15], which propose to use the marginal empirical characteristic
function to estimate the one-dimensional parameter of the kernel function and
the stability index of the driving L\'evy motion. We extend their work to allow
for a multi-parametric framework that in particular includes the important
examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck
process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a
periodic component among other models. We present both the consistency and the
associated central limit theorem of the minimal contrast estimator.
Furthermore, we demonstrate numerical analysis to uncover the finite sample
performance of our method.
