In this paper, we present two methods, induction and restriction procedures,
to construct new stable equivalences of Morita type. Suppose that a stable
equivalence of Morita type between two algebras $A$ and $B$ is defined by a
$B$-$A$-bimodule $N$. Then, for any finite admissible set $\Phi$ and any
generator $X$ of the $A$-module category, the $\Phi$-Auslander-Yoneda algebras
of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under
certain conditions, we transfer stable equivalences of Morita type between $A$
and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idempotent
elements in $A$ and $B$, respectively. Consequently, for self-injective
algebras $A$ and $B$ over a field without semisimple direct summands, and for
any $A$-module $X$ and $B$-module $Y$, if the $\Phi$-Auslander-Yoneda algebras
of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one
finite admissible set $\Phi$, then so are the $\Psi$-Auslander-Yoneda algebras
of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $\Psi$.
Moreover, two representation-finite algebras over a field without semisimple
direct summands are stably equivalent of Morita type if and only if so are
their Auslander algebras. As another consequence, we construct an infinite
family of algebras of the same dimension and the same dominant dimension such
that they are pairwise derived equivalent, but not stably equivalent of Morita
type. This answers a question by Thorsten Holm.