Optimal model averaging for single-index models with divergent dimensions

Abstract: 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.