  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 $\mathrm{II}_1$ 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 $\mathrm{II}_1$ factors is also discussed and some partial results are
given.
