  The ill-posedness of the inverse problem of recovering a regression function
in a nonparametric instrumental variable model leads to estimators that may
suffer from a very slow, logarithmic rate of convergence. In this paper, we
show that restricting the problem to models with monotone regression functions
and monotone instruments significantly weakens the ill-posedness of the
problem. In stark contrast to the existing literature, the presence of a
monotone instrument implies boundedness of our measure of ill-posedness when
restricted to the space of monotone functions. Based on this result we derive a
novel non-asymptotic error bound for the constrained estimator that imposes
monotonicity of the regression function. For a given sample size, the bound is
independent of the degree of ill-posedness as long as the regression function
is not too steep. As an implication, the bound allows us to show that the
constrained estimator converges at a fast, polynomial rate, independently of
the degree of ill-posedness, in a large, but slowly shrinking neighborhood of
constant functions. Our simulation study demonstrates significant finite-sample
performance gains from imposing monotonicity even when the regression function
is rather far from being a constant. We apply the constrained estimator to the
problem of estimating gasoline demand functions from U.S. data.
