  The reduced-rank vector autoregressive (VAR) model can be interpreted as a
supervised factor model, where two factor modelings are simultaneously applied
to response and predictor spaces. This article introduces a new model, called
vector autoregression with common response and predictor factors, to explore
further the common structure between the response and predictors in the VAR
framework. The new model can provide better physical interpretations and
improve estimation efficiency. In conjunction with the tensor operation, the
model can easily be extended to any finite-order VAR model. A
regularization-based method is considered for the high-dimensional estimation
with the gradient descent algorithm, and its computational and statistical
convergence guarantees are established. For data with pervasive cross-sectional
dependence, a transformation for responses is developed to alleviate the
diverging eigenvalue effect. Moreover, we consider additional sparsity
structure in factor loading for the case of ultra-high dimension. Simulation
experiments confirm our theoretical findings and a macroeconomic application
showcases the appealing properties of the proposed model in structural analysis
and forecasting.
