  The double ramification hierarchy is a new integrable hierarchy of
hamiltonian PDEs introduced recently by the first author. It is associated to
an arbitrary given cohomological field theory. In this paper we study the
double ramification hierarchy associated to the cohomological field theory
formed by Witten's $r$-spin classes. Using the formula for the product of the
top Chern class of the Hodge bundle with Witten's class, found by the second
author, we present an effective method for a computation of the double
ramification hierarchy. We do explicit computations for $r=3,4,5$ and prove
that the double ramification hierarchy is Miura equivalent to the corresponding
Dubrovin--Zhang hierarchy. As an application, this result together with a
recent work of the first author with Paolo Rossi gives a quantization of the
$r$-th Gelfand--Dickey hierarchy for $r=3,4,5$.
