Partial Envelope and Reduced-Rank Partial Envelope Vector Autoregressive Models

Traditional vector autoregressive (VAR) models have been extensively used in modeling multivariate time series data due to their flexibility and simplicity. However, they often suffer from overparameterization, particularly in high-dimensional datasets, limiting the inclusion of variables and lags. In many cases, specific lag effects may be of greater interest, particularly when significant cross-correlations occur at shorter lags. To address these challenges, we first propose the partial envelope VAR (PEVAR), which identifies and removes white noise and immaterial components of complex time series data by linking the mean function with covariance structures via a reduced subspace. The PEVAR model concentrates on a subset of important lag variables to improve estimation efficiency. We further extend this approach by integrating the partial envelope concept into the reduced-rank VAR framework and propose the reduced-rank partial envelope VAR (RPEVAR) model. RPEVAR simultaneously integrates the strengths of both the PEVAR and partial reduced-rank VAR models. These models offer a parsimonious technique by focusing on relevant lag variables during coefficient estimation, potentially leading to significant efficiency gains and enhanced accuracy relative to non-partial VAR models. Supplementary materials for this article are available online.