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The reduced-rank vector autoregressive (VAR) model can be interpreted as asupervised factor model, where two factor modelings are simultaneously appliedto response and predictor spaces. This article introduces a new model, calledvector autoregression with common response and predictor factors, to explorefurther the common structure between the response and predictors in the VARframework. The new model can provide better physical interpretations andimprove estimation efficiency. In conjunction with the tensor operation, themodel can easily be extended to any finite-order VAR model. Aregularization-based method is considered for the high-dimensional estimationwith the gradient descent algorithm, and its computational and statisticalconvergence guarantees are established. For data with pervasive cross-sectionaldependence, a transformation for responses is developed to alleviate thediverging eigenvalue effect. Moreover, we consider additional sparsitystructure in factor loading for the case of ultra-high dimension. Simulationexperiments confirm our theoretical findings and a macroeconomic applicationshowcases the appealing properties of the proposed model in structural analysisand forecasting.

High-Dimensional Vector Autoregression with Common Response and Predictor Factors