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

Towards a description of the double ramification hierarchy for Witten's $r$-spin class