  We consider the estimation of a structural function which models a
non-parametric relationship between a response and an endogenous regressor
given an instrument in presence of dependence in the data generating process.
Assuming an independent and identically distributed (iid.) sample it has been
shown in Johannes and Schwarz (2010) that a least squares estimator based on
dimension reduction and thresholding can attain minimax-optimal rates of
convergence up to a constant. As this estimation procedure requires an optimal
choice of a dimension parameter with regard amongst others to certain
characteristics of the unknown structural function we investigate its fully
data-driven choice based on a combination of model selection and Lepski's
method inspired by Goldenshluger and Lepski (2011). For the resulting fully
data-driven thresholded least squares estimator a non-asymptotic oracle risk
bound is derived by considering either an iid. sample or by dismissing the
independence assumption. In both cases the derived risk bounds coincide up to a
constant assuming sufficiently weak dependence characterised by a fast decay of
the mixing coefficients. Employing the risk bounds the minimax optimality up to
constant of the estimator is established over a variety of classes of
structural functions.
