Finite-time analysis of vector autoregressive models under linear restrictions
成果类型：
Article
署名作者：
Zheng, Yao; Cheng, Guang
署名单位：
University of Connecticut; Purdue University System; Purdue University
刊物名称：
BIOMETRIKA
ISSN/ISSBN：
0006-3444
DOI：
10.1093/biomet/asaa065
发表日期：
2021
页码：
469489
关键词：
摘要：
This paper develops a unified finite-time theory for the ordinary least squares estimation of possibly unstable and even slightly explosive vector autoregressive models under linear restrictions, with the applicable region rho(A) <= 1 + c/n, where rho(A) is the spectral radius of the transition matrix A in the VAR(1) representation, n is the time horizon and c > 0 is a universal constant. The linear restriction framework encompasses various existing models such as banded/network vector autoregressive models. We show that the restrictions reduce the error bounds via not only the reduced dimensionality, but also a scale factor resembling the asymptotic covariance matrix of the estimator in the fixed-dimensional set-up: as long as the model is correctly specified, this scale factor is decreasing in the number of restrictions. It is revealed that the phase transition from slow to fast error rate regimes is determined by the smallest singular value of A, a measure of the least excitable mode of the system. The minimax lower bounds are derived across different regimes. The developed non-asymptotic theory not only bridges the theoretical gap between stable and unstable regimes, but precisely characterizes the effect of restrictions and its interplay with model parameters. Simulations support our theoretical results.
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