  Muhly and Solel developed a notion of Morita equivalence for $C^{*}$-
correspondences, which they used to show that if two $C^{*}$-correspondences
$E$ and $F$ are Morita equivalent then their tensor algebras
$\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent
operator algebras. We give the weak$^{*}$ version of this result by considering
(weak) Morita equivalence of $W^{*}$-correspondences and employing Blecher and
Kashyap's notion of Morita equivalence for dual operator algebras. More
precisely, we show that weak Morita equivalence of $W^{*}$-correspondences $E$
and $F$ implies weak Morita equivalence of their Hardy algebras $H^{\infty}(E)$
and $H^{\infty}(F)$. We give special attention to $W^{*}$-graph correspondences
and show a number of results related to their Morita equivalence.
