  We provide finite sample properties of sparse multivariate ARCH processes,
where the linear representation of ARCH models allows for an ordinary least
squares estimation. Under the restricted strong convexity of the unpenalized
loss function, regularity conditions on the penalty function, strict stationary
and beta-mixing process, we prove non-asymptotic error bounds on the
regularized ARCH estimator. Based on the primal-dual witness method of Loh and
Wainwright (2017), we establish variable selection consistency, including the
case when the penalty function is non-convex. These theoretical results are
supported by empirical studies.
