Stability of products of equivalence relations

An ergodic p.m.p. equivalence relation $\mathcal{R}$ is said to be stable if $\mathcal{R}\cong \mathcal{R}\times {\mathcal{R}}_0$ where ${\mathcal{R}}_0$ is the unique hyperfinite ergodic type ${\mathrm{I}\mathrm{I}}_1$ equivalence relation. We prove that a direct product $\mathcal{R}\times \mathcal{S}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components $\mathcal{R}$ or $\mathcal{S}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff ${\mathrm{I}\mathrm{I}}_1$ factors is also discussed and some partial results are given.