##
**Power-law partial correlation network models.**
*(English)*
Zbl 1403.62093

Summary: 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.

### MSC:

62H12 | Estimation in multivariate analysis |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

05C80 | Random graphs (graph-theoretic aspects) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

### Software:

plfit
PDF
BibTeX
XML
Cite

\textit{M. Barigozzi} et al., Electron. J. Stat. 12, No. 2, 2905--2929 (2018; Zbl 1403.62093)

### References:

[1] | Acemoglu, D., Carvalho, V., Ozdaglar, A., and Tahbaz-Salehi, A. (2012). The Network Origins of Aggregate Fluctuations., Econometrica, 80, 1977–2016. · Zbl 1274.91317 |

[2] | Anderson, W. and Morley, T. (1985). Eigenvalues of the Laplacian of a graph., Linear and Multilinear Algebra, 18, 141–145. · Zbl 0594.05046 |

[3] | Bai, J. (2003). Inferential Theory for Factor Models of Large Dimensions., Econometrica, 71, 135–171. · Zbl 1136.62354 |

[4] | Bickel, P. J. and Levina, E. (2008). High Dimensional Inference and Random Matrices., The Annals of Statistics, 36, 2577–2604. |

[5] | Bollobás, B., Janson, S., and Riordan, O. (2007). The phase transition in inhomogeneous random graphs., Random Structures & Algorithms, 31, 3–122. · Zbl 1123.05083 |

[6] | 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. · Zbl 1144.05315 |

[7] | Brouwer, A. E. and Haemers, W. (2011)., Spectra of graphs. Springer. · Zbl 1231.05001 |

[8] | Brownlees, C., Nualart, E., and Sun, Y. (2018). Realized Networks., Journal of Applied Econometrics, fothcoming. |

[9] | Chamberlain, G. and Rothschild, M. (1983). Arbitrage, factor structure, and mean-variance analysis on large asset markets., Econometrica, 51, 1281–1304. · Zbl 0523.90017 |

[10] | Chung, F. (1997)., Spectral graph theory, volume 92. American Mathematical Soc. · Zbl 0867.05046 |

[11] | Chung, F. and Lu, L. (2006)., Complex Graphs and Networks. American Mathematical Society, Providence. · Zbl 1114.90071 |

[12] | Clauset, A., Shalizi, C. R., and Newman, M. E. (2009). Power-law distributions in empirical data., SIAM review, 51, 661–703. · Zbl 1176.62001 |

[13] | 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. |

[14] | Dempster, A. P. (1972). Covariance selection., Biometrics, 28, 157–175. |

[15] | 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. · Zbl 1311.91196 |

[16] | 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. |

[17] | Hagerup, T. and Rüb, C. (1990). A guided tour of Chernoff bounds., Information Processing Letters, 33, 305–308. · Zbl 0702.60021 |

[18] | 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. · Zbl 0521.62024 |

[19] | 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. |

[20] | Hautsch, N., Schaumburg, J., and Schienle, M. (2014a). Financial network systemic risk contributions., Review of Finance. available online. · Zbl 1417.91560 |

[21] | Hautsch, N., Schaumburg, J., and Schienle, M. (2014b). Forecasting systemic impact in financial networks., International Journal of Forecasting, 30, 781–794. |

[22] | 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. · Zbl 0323.62033 |

[23] | Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables., Journal of the American Statistical Association, 58, 13–30. · Zbl 0127.10602 |

[24] | Karp, R. (1988)., Probabilistic Analysis of Algorithms. Class Notes, University of California, Berkeley. |

[25] | Lam, C. (2016). Nonparametric eigenvalue-regularized precision or covariance matrix estimator., The Annals of Statistics, 44, 928–953. · Zbl 1341.62124 |

[26] | Lam, C. and Fan, J. (2009). Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation., The Annals of Statistics, 37, 4254–4278. · Zbl 1191.62101 |

[27] | Lauritzen, S. L. (1996)., Graphical Models. Clarendon Press, Oxford. · Zbl 0907.62001 |

[28] | Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices., Journal of multivariate analysis, 88, 365–411. · Zbl 1032.62050 |

[29] | 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. · Zbl 1388.62046 |

[30] | Pourahmadi, M. (2011). Covariance Estimation: The GLM and the Regularization Perspectives., Statistical Science, 26, 369–387. · Zbl 1246.62139 |

[31] | Ross, S. A., et al. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13, 341–360. |

[32] | 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. · Zbl 1041.62081 |

[33] | Stock, J. H. and Watson, M. W. (2002b). Macroeconomic Forecasting Using Diffusion Indexes., Journal of Business and Economic Statistics, 20, 147–162. |

[34] | van der Hofstad, R. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.