  We suggest two nonparametric approaches, based on kernel methods and
orthogonal series to estimating regression functions in the presence of
instrumental variables. For the first time in this class of problems, we derive
optimal convergence rates, and show that they are attained by particular
estimators. In the presence of instrumental variables the relation that
identifies the regression function also defines an ill-posed inverse problem,
the ``difficulty'' of which depends on eigenvalues of a certain integral
operator which is determined by the joint density of endogenous and
instrumental variables. We delineate the role played by problem difficulty in
determining both the optimal convergence rate and the appropriate choice of
smoothing parameter.
