  In this paper we investigate the reverse mathematics of higher-order
analogues of the theory \ATRz{} within the framework of higher order reverse
mathematics developed by Kohlenbach \cite{Koh01}. We define a theory \RCAzthr,
a close higher-type analogue of the classical base theory \RCAz, and show that
it is essentially a conservative subtheory of Kohlenbach's base theory \RCAzo.
Working over \RCAzthr, we study higher-type analogues of statements classically
equivalent to \ATRz, including open and clopen determinacy, as well as two
choice principles, and prove several equivalences and separations. Our main
result is the separation of open and clopen determinacy for reals, using a
variant of Steel forcing; in the presentation of this result, we develop a new,
more flexible framework for Steel-type forcing.
