  We consider the problem of estimating the slope parameter in functional
linear instrumental regression, where in the presence of an instrument W, i.e.,
an exogenous random function, a scalar response Y is modeled in dependence of
an endogenous random function X. Assuming second order stationarity jointly for
X and W a nonparametric estimator of the functional slope parameter and its
derivatives is proposed based on an n-sample of (Y,X,W). In this paper the
minimax optimal rate of convergence of the estimator is derived assuming that
the slope parameter belongs to the well-known Sobolev space of periodic
functions. We discuss the cases that the cross-covariance operator associated
to the random functions X and W is finitely, infinitely or in some general form
smoothing.
