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.
Highlights
The recent financial crises in the United States and Europe have boosted the interest in network analysis in economics and finance
In this last section we carry out an empirical study to assess to which extent real data presents empirical features that are consistent with the power-law partial correlation network model
In this work we introduce a tractable class of partial correlation network models whose underlying network structure is a function of a random graph
Summary
The recent financial crises in the United States and Europe have boosted the interest in network analysis in economics and finance. In the aftermath of these crises, several authors have proposed network estimation techniques for large panels of time series These methods are typically applied to carry out inference on the degree of interdependence among firms on the basis of market data, such as stock prices. The largest eigenvalues of the concentration matrix can be used to learn the power-law tail parameter of the random graph, which characterizes the power-law structure of the network This allows us to carry out inference on the degree of interdependence in the system without having to estimate the underlying partial correlation structure of the data (typically, using LASSO as in Peng et al [30]), which can be computationally challenging when the system dimensionality is large.
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