  Graphical continuous Lyapunov models offer a new perspective on modeling
causally interpretable dependence structure in multivariate data by treating
each independent observation as a one-time cross-sectional snapshot of a
temporal process. Specifically, the models assume that the observations are
cross-sections of independent multivariate Ornstein-Uhlenbeck processes in
equilibrium. The Gaussian equilibrium exists under a stability assumption on
the drift matrix, and the equilibrium covariance matrix is determined by the
continuous Lyapunov equation. Each graphical continuous Lyapunov model assumes
the drift matrix to be sparse, with a support determined by a directed graph. A
natural approach to model selection in this setting is to use an
$\ell_1$-regularization technique that, based on a given sample covariance
matrix, seeks to find a sparse approximate solution to the Lyapunov equation.
We study the model selection properties of the resulting lasso technique to
arrive at a consistency result. Our detailed analysis reveals that the involved
irrepresentability condition is surprisingly difficult to satisfy. While this
may prevent asymptotic consistency in model selection, our numerical
experiments indicate that even if the theoretical requirements for consistency
are not met, the lasso approach is able to recover relevant structure of the
drift matrix and is robust to aspects of model misspecification.
