Factorized estimation of high‐dimensional nonparametric covariance models

Estimation of covariate‐dependent conditional covariance matrix in a high‐dimensional space poses a challenge to contemporary statistical research. The existing kernel estimators may not be locally adaptive due to using a single bandwidth to explore the smoothness of all entries of the target matrix function. In this paper, we propose a novel framework to address this issue, where we factorize the target matrix into factors and estimate these factors in turn by the kernel approach. The resulting estimator is further regularized by thresholding and optimal shrinkage. Under certain mixing and sparsity conditions, we show that the proposed estimator is well‐conditioned and uniformly consistent with the underlying matrix function even when the sample is dependent. Simulation studies suggest that the proposed estimator significantly outperforms its competitors in terms of integrated root‐squared estimation error. We present an application to financial return data.