Ripples on financial networks

Abstract

In the financial markets, asset returns exhibit collective dynamics masking individual impacts on the rest of the market. Hence, it is still an open problem to identify how shocks originating from one particular asset create spillover effects across other assets. The problem is more acute when there is a large number of simultaneously traded assets, making the identification of which asset affects which other assets even more difficult. In this paper, we construct a network of the conditional volatility series estimated from asset returns and estimate a many-dimensional VAR model with unique identification criteria based on the network topology. Because of the interlinkages across stocks, volatility shock to a particular asset propagates through the network creating a ripple effect. Our method allows us to find the exact path the ripple effect follows on the whole network of assets.

Keywords:

• Volatility spillover
• VAR identification
• spectral analysis
• core-periphery

JEL CLASSIFICATIONS:

• C55
• C58
• G17
• C53

Acknowledgments

This research was partially supported by institute grant, IIM Ahmedabad. We wish to thank conference and seminar participants at the WEHIA'19, ICU Tokyo, Reserve Bank of India (Strategic Research Unit), IIT Kanpur, IIT Delhi, Dibrugarh University, IIM Calcutta, two anonymous reviewers and Giuseppe Brandi for useful comments. All remaining errors are ours.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 All data (prices as well as other company-level information) have been collected from Thomson Reuters Eikon database. See Tables 4 in the Appendix for stock names and ticker symbols.

2 We note that this step can be conducted in multiple ways. Notably, Diebold and Yilmaz (2015a) have popularized a formula due to Garman and Klass (1980) that utilizes daily highest prices, lowest prices, opening prices and closing prices, all measured in logs for all stocks. Because of the substantially lower informational requirement and the fact that our fundamental objective is to characterize spillover from one return series to another, we use GARCH methodology because of its simplicity.

3 See Table 1 in the Online Supplementary Material for ARCH-LM test on the return series for all the stocks and ADF test results for the conditional volatility series. For the GARCH estimation, we have used sgarch module in rugarch package in R.

4 For a given network $G=(V,E)$ with adjacency matrix A, we define eigenvector centrality to be a vector $c^{eig}$ which solves $A{c}^{eig}=\lambda c^{eig}$ where λ is chosen to be the maximum eigenvalue ${\lambda }_{max}$ of the adjacency matrix A. For the purpose of finding eigenvector centrality on the correlation matrix, we consider $A=\mathrm{\Gamma }$.

5 There are other centrality measures like degree, betweenness and closeness, etc. Here, we do not utilize them for three reasons. One, they are more useful for binary networks than weighted networks. Two, eigenvector centrality has algebraic properties that these measures do not possess. We have utilized these properties throughout the analysis. Three, eigenvector centrality has an interpretation in terms of market mode due to principal component analysis (see Section 5). Other centrality measures do not have this nice property.

6 Here y is a generic variable. For actual estimation, we substitute $y_{it}$ by $\mathrm{l}\mathrm{o}\mathrm{g}\left({\sigma }_{it}\right)$. Taking <roman > log < /roman > is important to take care of the skewness of the latent volatility process.

7 Here we note two points. First, since we are using MST only for the purpose of visualization (all computation and estimation have been carried out on the whole network), our results do not depend in any way on the MST. Second, there could be other approaches to reduce the network size. However, MST by construction retains exactly N−1 edges for N nodes and still retain the connectivity of the network. Therefore, MST would produce the smallest non-trivial network. Since for visualization, our goal is to present as clean a picture as possible, we have implemented the MST.

8 There are many standard algorithms to create MST from a fully connected graph. Prim's and Kruskal's algorithms are standard ones. Also, MST is unique for a graph when all edges have distinct weights (which is the case for an adjacency matrix obtained from correlation matrix). Interested readers can refer to Kreyszig (2015) for a textbook description of the minimum spanning tree and the relevant methodology used here.

9 In this paper, we have analyzed daily data.

10 Full network with all connections is too dense to visually identify the nodes, edges and how shocks are propagating.

11 Complete description of the company names and the corresponding short forms (ticker symbols) can be found in the Appendix. See Table 4 for the listed companies considered in this paper.

12 We thank a reviewer for this suggestion. See Figure 1 in the Online Supplementary Material for a frequency distribution of all the differences in magnitude between the estimated impulse response functions and their simulated counterparts. The differences are minimal.

13 See Tables 2 and 3 in the Online Supplementary Material for a rolling window estimation with yearly overlaps from 2002–05 to 2014–17.

14 Indegree represents how much a particular node in a graph is open to influence of other nodes and outdegree represents direct influence of one particular node on the remaining nodes.

Additional information

Funding

This work was supported by Indian Institute of Management Ahmedabad [R&P Grant].