Electronic Journal of Statistics

Power-law partial correlation network models

Matteo Barigozzi, Christian Brownlees, and Gábor Lugosi

Full-text: Open access

Abstract

We introduce a class of partial correlation network models whose network structure is determined by a random graph. In particular in this work we focus on a version of the model in which the random graph has a power-law degree distribution. A number of cross-sectional dependence properties of this class of models are derived. The main result we establish is that when the random graph is power-law, the system exhibits a high degree of collinearity. More precisely, the largest eigenvalues of the inverse covariance matrix converge to an affine function of the degrees of the most interconnected vertices in the network. The result implies that the largest eigenvalues of the inverse covariance matrix are approximately power-law distributed, and that, as the system dimension increases, the eigenvalues diverge. As an empirical illustration we analyse two panels of stock returns of companies listed in the S&P 500 and S&P 1500 and show that the covariance matrices of returns exhibits empirical features that are consistent with our power-law model.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 2905-2929.

Dates
Received: May 2017
First available in Project Euclid: 18 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1537257629

Digital Object Identifier
doi:10.1214/18-EJS1478

Keywords
Partial correlation networks random graphs power-law

Rights
Creative Commons Attribution 4.0 International License.

Citation

Barigozzi, Matteo; Brownlees, Christian; Lugosi, Gábor. Power-law partial correlation network models. Electron. J. Statist. 12 (2018), no. 2, 2905--2929. doi:10.1214/18-EJS1478. https://projecteuclid.org/euclid.ejs/1537257629


Export citation

References

  • [1] Acemoglu, D., Carvalho, V., Ozdaglar, A., and Tahbaz-Salehi, A. (2012). The Network Origins of Aggregate Fluctuations., Econometrica, 80, 1977–2016.
  • [2] Anderson, W. and Morley, T. (1985). Eigenvalues of the Laplacian of a graph., Linear and Multilinear Algebra, 18, 141–145.
  • [3] Bai, J. (2003). Inferential Theory for Factor Models of Large Dimensions., Econometrica, 71, 135–171.
  • [4] Barigozzi, M. and Brownlees, C. (2016). NETS: Network Estimation for Time Series. Technical report, Barcelona, GSE.
  • [5] Bickel, P. J. and Levina, E. (2008). High Dimensional Inference and Random Matrices., The Annals of Statistics, 36, 2577–2604.
  • [6] Bollobás, B., Janson, S., and Riordan, O. (2007). The phase transition in inhomogeneous random graphs., Random Structures & Algorithms, 31, 3–122.
  • [7] Brouwer, A. E. and Haemers, W. (2008). A lower bound for the laplacian eigenvalues of a graph-proof of a conjecture by guo., Linear Algebra and Applications, 429, 2131–2135.
  • [8] Brouwer, A. E. and Haemers, W. (2011)., Spectra of graphs. Springer.
  • [9] Brownlees, C., Nualart, E., and Sun, Y. (2018). Realized Networks., Journal of Applied Econometrics, fothcoming.
  • [10] Chamberlain, G. and Rothschild, M. (1983). Arbitrage, factor structure, and mean-variance analysis on large asset markets., Econometrica, 51, 1281–1304.
  • [11] Chung, F. (1997)., Spectral graph theory, volume 92. American Mathematical Soc.
  • [12] Chung, F. and Lu, L. (2006)., Complex Graphs and Networks. American Mathematical Society, Providence.
  • [13] Clauset, A., Shalizi, C. R., and Newman, M. E. (2009). Power-law distributions in empirical data., SIAM review, 51, 661–703.
  • [14] Connor, G. and Korajczyk, R. A. (1993). A test for the number of factors in an approximate factor model., The Journal of Finance, 48, 1263–1291.
  • [15] Dempster, A. P. (1972). Covariance selection., Biometrics, 28, 157–175.
  • [16] Diebold, F. X. and Yılmaz, K. (2014). On the network topology of variance decompositions: Measuring the connectedness of financial firms., Journal of Econometrics, 182, 119–134.
  • [17] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs., Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.
  • [18] Hagerup, T. and Rüb, C. (1990). A guided tour of Chernoff bounds., Information Processing Letters, 33, 305–308.
  • [19] Hall, P. (1982). On some simple estimates of an exponent of regular variation., Journal of the Royal Statistical Society. Series B (Methodological), pages 37–42.
  • [20] Hartmann, P., Straetmans, S., and de Vries, C. (2007). Banking System Stability. A Cross-Atlantic Perspective. In, The Risks of Financial Institutions, NBER Chapters, pages 133–192. National Bureau of Economic Research, Inc.
  • [21] Hautsch, N., Schaumburg, J., and Schienle, M. (2014a). Financial network systemic risk contributions., Review of Finance. available online.
  • [22] Hautsch, N., Schaumburg, J., and Schienle, M. (2014b). Forecasting systemic impact in financial networks., International Journal of Forecasting, 30, 781–794.
  • [23] Hill, B. M., et al. (1975). A simple general approach to inference about the tail of a distribution. The annals of statistics, 3, 1163–1174.
  • [24] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables., Journal of the American Statistical Association, 58, 13–30.
  • [25] Karp, R. (1988)., Probabilistic Analysis of Algorithms. Class Notes, University of California, Berkeley.
  • [26] Lam, C. (2016). Nonparametric eigenvalue-regularized precision or covariance matrix estimator., The Annals of Statistics, 44, 928–953.
  • [27] Lam, C. and Fan, J. (2009). Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation., The Annals of Statistics, 37, 4254–4278.
  • [28] Lauritzen, S. L. (1996)., Graphical Models. Clarendon Press, Oxford.
  • [29] Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices., Journal of multivariate analysis, 88, 365–411.
  • [30] Peng, J., Wang, P., Zhou, N., and Zhu, J. (2009). Partial Correlation Estimation by Joint Sparse Regression Models., Journal of the American Statistical Association, 104, 735–746.
  • [31] Pourahmadi, M. (2011). Covariance Estimation: The GLM and the Regularization Perspectives., Statistical Science, 26, 369–387.
  • [32] Ross, S. A., et al. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13, 341–360.
  • [33] Stock, J. H. and Watson, M. W. (2002a). Forecasting using principal components from a large number of predictors., Journal of the American Statistical Association, 97, 1167–1179.
  • [34] Stock, J. H. and Watson, M. W. (2002b). Macroeconomic Forecasting Using Diffusion Indexes., Journal of Business and Economic Statistics, 20, 147–162.
  • [35] van der Hofstad, R. (2015)., Random Graphs and Complex Networks.