Convolution transformation for sufficient dimension reduction in time series

Abstract While univariate time series analysis is well established, capturing nonlinear dynamics and high-dimensional lag structure remains a significant challenge. Sufficient dimension reduction (SDR) provides a useful and powerful framework to address the curse of dimensionality in such contexts. In this study, we introduce a convolution transformation method (CM) for estimating SDR subspaces in time series, with particular focus on the time series central mean subspace (TS-CMS) and the time series central subspace (TS-CS). We develop explicit matrix-based estimators for these two subspaces and extend a Fourier transform-based method (FM), originally developed for TS-CMS estimation, to the broader TS-CS setting. The asymptotic properties of the proposed estimators are established. Through simulation studies and empirical applications, we compare the performance of the CM and FM estimators. The numerical results suggest that the CM estimator performs well for estimating the TS-CMS, while the Fourier transform-based procedure tends to perform better for estimating the TS-CS in the settings considered.