Before revision:  We show that in many cases the best approximation coincides with the structural parameter and can be a useful and tractable object to infer relation between structural variables otherwise. Unlike in the identified case, estimation of the operator may have a non-negligible impact on the estimation accuracy and inference. We develop inferential methods for linear functionals in non-identified models as well as honest uniform confidence sets for the best approximation.
After revision:  We show that in many cases the best approximation coincides with the structural parameter or at least may reasonably approximate it. We characterize the nonasymptotic Hilbert space norm and the uniform norm convergence rates for the best approximation.