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In this paper, we consider the nonparametric smoothing technique with both discrete and categorical regressors. In the existing literature, it is generally admitted that it is better to smooth the discrete variables, which is similar to the smoothing technique for continuous regressors but using discrete kernels. However, as we explain in this paper, such approach might lead to a potential problem which is linked to the bandwidth selection for the continuous regressors due to the presence of the discrete regressors. Through the numerical study, we find that in many cases, the performance of the resulting nonparametric regression estimates may deteriorate if the discrete variables are smoothed in the way addressed so far in the literature, and that a fully separate estimation without any smoothing of the discrete variables may provide significantly better results. As a solution, we suggest a simple generalization of the popular approach proposed by Racine and Li [Journal of Econometrics, 2004] to address this problem and to provide estimates with more robust performance. We analyze the problem theoretically, develop the asymptotic theory for the new nonparametric smoothing method and study the finite sample behavior of the proposed generalized approach through extensive Monte-Carlo experiments as well present an empirical illustration.

This article studies multiple structural breaks in large contemporaneous covariance matrices of high-dimensional time series satisfying an approximate factor model. The breaks in the second-order moment structure of the common components are due to sudden changes in either factor loadings or covariance of latent factors, requiring appropriate transformation of the factor models to facilitate estimation of the (transformed) common factors and factor loadings via the classical principal component analysis. With the estimated factors and idiosyncratic errors, an easy-to-implement CUSUM-based detection technique is introduced to consistently estimate the location and number of breaks and correctly identify whether they originate in the common or idiosyncratic error components. The algorithms of Wild Binary Segmentation for Covariance (WBS-Cov) and Wild Sparsified Binary Segmentation for Covariance (WSBS-Cov) are used to estimate breaks in the common and idiosyncratic error components, respectively. Under some technical conditions, the asymptotic properties of the proposed methodology are derived with near-optimal rates (up to a logarithmic factor) achieved for the estimated breaks. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the developed method and its comparison with other existing approaches. We finally apply our method to study the contemporaneous covariance structure of daily returns of S&P 500 constituents and identify a few breaks including those occurring during the 2007–2008 financial crisis and the recent coronavirus (COVID-19) outbreak. An R package “BSCOV” is provided to implement the proposed algorithms. Supplementary materials for this article are available online.

We study the estimation of conditional mean regression functions through the so-called subset-based kernel principal component analysis (KPCA). Instead of using one global kernel feature space, we project a target function into different localized kernel feature spaces at different parts of the sample space. Each localized kernel feature space reflects the relationship on a subset between the response and covariates more parsimoniously. When the observations are collected from a strictly stationary and weakly dependent process, the orthonormal eigenfunctions which span the kernel feature space are consistently estimated by implementing an eigenanalysis on the subset-based kernel Gram matrix, and the estimated eigenfunctions are then used to construct the estimation of the mean regression function. Under some regularity conditions, the developed estimator is shown to be uniformly consistent over the subset with a convergence rate faster than those of some well-known nonparametric estimation methods. In addition, we also discuss some generalizations of the KPCA approach, and consider using the same subset-based KPCA approach to estimate the conditional distribution function. The numerical studies including three simulated examples and two real data sets illustrate the reliable performance of the proposed method. In particular, the improvement over the global KPCA method is evident.

Local linear fitting is a popular nonparametric method in nonlinear statistical and econometric modelling. Lu and Linton (2007) established the point wise asymptotic distribution (central limit theorem) for the local linear estimator of nonparametric regression function under the condition of near epoch dependence. We further investigate the uniform consistency of this estimator. The uniformly strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results of uniform convergence rates for nonparametric kernel-based estimators are provided. Applications of our results to conditional variance function estimation and some economic time series models are also discussed. The results of this paper will be of widely potential interest in time series semiparametric modelling.