This paper considers estimation and model selection of quantile vector autoregression (QVAR). Conventional quantile regression often yields undesirable crossing quantile curves, violating the monotonicity of quantiles. To address this issue, we propose a simplex quantile vector autoregression (SQVAR) framework, which transforms the autoregressive (AR) structure of the original QVAR model into a simplex, ensuring that the estimated quantile curves remain monotonic across all quantile levels. In addition, we impose the smoothly clipped absolute deviation (SCAD) penalty on the SQVAR model to mitigate the explosive nature of the parameter space. We further develop a Bayesian information criterion (BIC)-based procedure for selecting the optimal penalty parameter and introduce new frameworks for impulse response analysis of QVAR models. Finally, we establish asymptotic properties of the proposed method, including the convergence rate and asymptotic normality of the estimator, the consistency of AR order selection, and the validity of the BIC-based penalty selection. For illustration, we apply the proposed method to U.S. stock market data, highlighting the usefulness of our SQVAR method.

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