Functional linear instrumental regression under second order stationarity

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.