  This paper offers a new approach to address the model uncertainty in
(potentially) divergent-dimensional single-index models (SIMs). We propose a
model-averaging estimator based on cross-validation, which allows the dimension
of covariates and the number of candidate models to increase with the sample
size. We show that when all candidate models are misspecified, our
model-averaging estimator is asymptotically optimal in the sense that its
squared loss is asymptotically identical to that of the infeasible best
possible averaging estimator. In a different situation where correct models are
available in the model set, the proposed weighting scheme assigns all weights
to the correct models in the asymptotic sense. We also extend our method to
average regularized estimators and propose pre-screening methods to deal with
cases with high-dimensional covariates. We illustrate the merits of our method
via simulations and two empirical applications.
