The Annals of Statistics

Network vector autoregression

Xuening Zhu, Rui Pan, Guodong Li, Yuewen Liu, and Hansheng Wang

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We consider here a large-scale social network with a continuous response observed for each node at equally spaced time points. The responses from different nodes constitute an ultra-high dimensional vector, whose time series dynamic is to be investigated. In addition, the network structure is also taken into consideration, for which we propose a network vector autoregressive (NAR) model. The NAR model assumes each node’s response at a given time point as a linear combination of (a) its previous value, (b) the average of its connected neighbors, (c) a set of node-specific covariates and (d) an independent noise. The corresponding coefficients are referred to as the momentum effect, the network effect and the nodal effect, respectively. Conditions for strict stationarity of the NAR models are obtained. In order to estimate the NAR model, an ordinary least squares type estimator is developed, and its asymptotic properties are investigated. We further illustrate the usefulness of the NAR model through a number of interesting potential applications. Simulation studies and an empirical example are presented.

Article information

Ann. Statist., Volume 45, Number 3 (2017), 1096-1123.

Received: August 2015
Revised: April 2016
First available in Project Euclid: 13 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62J05: Linear regression

Multivariate time series ordinary least squares social network vector autoregression


Zhu, Xuening; Pan, Rui; Li, Guodong; Liu, Yuewen; Wang, Hansheng. Network vector autoregression. Ann. Statist. 45 (2017), no. 3, 1096--1123. doi:10.1214/16-AOS1476.

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Supplemental materials

  • Supplement to “Network vector autoregression”. The supplementary material [35] contains the verification of (2.6) and (2.7), proofs of Theorem 1, Theorem 4, Theorem 5, two useful lemmas and Proposition 2. The numerical verification of conditions (C2)–(C3) are also included.