  Composite likelihood has shown promise in settings where the number of
parameters $p$ is large due to its ability to break down complex models into
simpler components, thus enabling inference even when the full likelihood is
not tractable. Although there are a number of ways to formulate a valid
composite likelihood in the finite-$p$ setting, there does not seem to exist
agreement on how to construct composite likelihoods that are comp utationally
efficient and statistically sound when $p$ is allowed to diverge. This article
introduces a method to select sparse composite likelihoods by minimizing a
criterion representing the statistical efficiency of the implied estimator plus
an $L_1$-penalty discouraging the inclusion of too many sub-likelihood terms.
Conditions under which consistent model selection occurs are studied. Examples
illustrating the procedure are analysed in detail and applied to real data.

Methodology