Scaled envelope models for multivariate time series

Vector autoregressive (VAR) models have become a popular choice for modeling multivariate time series data due to their simplicity and ease of use. Efficient estimation of VAR coefficients is an important problem. The envelope technique for VAR models is demonstrated to have the potential to yield significant gains in efficiency and accuracy by incorporating linear combinations of the response vector that are essentially immaterial to the estimation of the VAR coefficients. However, inferences based on envelope VAR (EVAR) models are not invariant or equivariant upon the rescaling of the VAR responses, limiting their application to time series data that are measured in the same or similar units. In scenarios where VAR responses are measured on different scales, the efficiency improvements promised by envelopes are not always guaranteed. To address this limitation, we introduce the scaled envelope VAR (SEVAR) model, which preserves the efficiency-boosting capabilities of standard envelope techniques while remaining invariant to scale changes. The asymptotic characteristics of the proposed estimators are established based on different error assumptions. Simulation studies and real-data analysis are conducted to demonstrate the efficiency and effectiveness of the proposed model. The numerical results corroborate our theoretical findings.