The Annals of Statistics

Network vector autoregression

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aos/1497319689

Digital Object Identifier
doi:10.1214/16-AOS1476

Mathematical Reviews number (MathSciNet)
MR3662449

Zentralblatt MATH identifier
1381.62256

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

Keywords
Multivariate time series ordinary least squares social network vector autoregression

Citation

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. https://projecteuclid.org/euclid.aos/1497319689


Export citation

References

  • [1] Anselin, L. (1999). Spatial Econometrics: Methods and Models. Springer Science and Business Media, Dordrecht.
  • [2] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • [3] Bedrick, E. J. and Tsai, C. L. (1994). Model selection for multivariate regression in small samples. Biometrics 50 226–231.
  • [4] Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd ed. Prentice Hall, Englewood Cliffs, NJ.
  • [5] Brown, L. D., Hagerman, R. L., Griffin, P. A. and Zmijewski, M. E. (1987). Security analyst superiority relative to univariate time-series models in forecasting quarterly earnings. J. Account. Econ. 9 61–87.
  • [6] Chen, X., Chen, Y. and Xiao, P. (2013). The impact of sampling and network topology on the estimation of social intercorrelations. J. Mark. Res. 50 95–110.
  • [7] Clauset, A., Shalizi, C. R. and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Rev. 51 661–703.
  • [8] Cuaresma, J. C., Hlouskova, J., Kossmeier, S. and Obersteiner, M. (2004). Forecasting electricity spot-prices using linear univariate time-series models. Appl. Energy 77 87–106.
  • [9] Dantzig, G. B. (1998). Linear Programming and Extensions, corrected ed. Princeton Univ. Press, Princeton, NJ.
  • [10] De Mol, C., Giannone, D. and Reichlin, L. (2008). Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components? J. Econometrics 146 318–328.
  • [11] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • [12] Fingleton, B. (1999). Spurious spatial regression: Some Monte Carlo results with a spatial unit root and spatial cointegration. J. Reg. Sci. 39 1–19.
  • [13] Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005). The generalized dynamic factor model: One-sided estimation and forecasting. J. Amer. Statist. Assoc. 100 830–840.
  • [14] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • [15] Hamilton, J. D. (1994). Time Series Analysis. Princeton Univ. Press, Princeton, NJ.
  • [16] Han, F. and Liu, H. (2013). Transition matrix estimation in high dimensional time series. In Proceedings of the 30th International Conference on Machine Learning (ICML-13) 172–180. ACM, Atlanta, GA.
  • [17] Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. J. Amer. Statist. Assoc. 76 33–65.
  • [18] Hsu, N.-J., Hung, H.-L. and Chang, Y.-M. (2008). Subset selection for vector autoregressive processes using Lasso. Comput. Statist. Data Anal. 52 3645–3657.
  • [19] Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist. 40 694–726.
  • [20] Lee, L.-F. (2004). Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72 1899–1925.
  • [21] Lee, L. F. and Yu, J. (2009). Spatial nonstationarity and spurious regression: The case with a row-normalized spatial weights matrix. Spatial Econ. Anal. 4 301–327.
  • [22] Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer, Berlin.
  • [23] McQuarrie, A. D. R. and Tsai, C.-L. (1998). Regression and Time Series Model Selection. World Scientific, River Edge, NJ.
  • [24] Negahban, S. and Wainwright, M. J. (2011). Estimation of (near) low-rank matrices with noise and high-dimensional scaling. Ann. Statist. 39 1069–1097.
  • [25] Newbold, P. and Granger, C. W. J. (1974). Experience with forecasting univariate time series and the combination of forecasts. J. Roy. Statist. Soc. Ser. A 137 131–164.
  • [26] Nowicki, K. and Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96 1077–1087.
  • [27] Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors. Biometrika 95 365–379.
  • [28] Park, B. U., Mammen, E., Härdle, W. and Borak, S. (2009). Time series modelling with semiparametric factor dynamics. J. Amer. Statist. Assoc. 104 284–298.
  • [29] Reinsel, G. C. and Velu, R. P. (1998). Multivariate Reduced-Rank Regression: Theory and Applications. Lecture Notes in Statistics 136. Springer, New York.
  • [30] Shumway, R. H. and Stoffer, D. S. (2000). Time Series Analysis and Its Applications. Springer, New York.
  • [31] Wang, Y. J. and Wong, G. Y. (1987). Stochastic blockmodels for directed graphs. J. Amer. Statist. Assoc. 82 8–19.
  • [32] Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge Univ. Press, London.
  • [33] Zhao, Y., Levina, E. and Zhu, J. (2012). Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Statist. 40 2266–2292.
  • [34] Zhou, J., Tu, Y., Chen, Y. and Wang, H. (2017). Estimating spatial autocorrelation with sampled network data. J. Bus. Econ. Stat. 35 130–138.
  • [35] Zhu, X., Pan, R., Li, G., Liu, Y. and Wang, H. (2016). Supplement to “Network vector autoregression.” DOI:10.1214/16-AOS1476SUPP.

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.