  Kernel maximum moment restriction (KMMR) recently emerges as a popular
framework for instrumental variable (IV) based conditional moment restriction
(CMR) models with important applications in conditional moment (CM) testing and
parameter estimation for IV regression and proximal causal learning. The
effectiveness of this framework, however, depends critically on the choice of a
reproducing kernel Hilbert space (RKHS) chosen as a space of instruments. In
this work, we presents a systematic way to select the instrument space for
parameter estimation based on a principle of the least identifiable instrument
space (LIIS) that identifies model parameters with the least space complexity.
Our selection criterion combines two distinct objectives to determine such an
optimal space: (i) a test criterion to check identifiability; (ii) an
information criterion based on the effective dimension of RKHSs as a complexity
measure. We analyze the consistency of our method in determining the LIIS, and
demonstrate its effectiveness for parameter estimation via simulations.
