# Statistical Modelling of Downside Risk Spillovers

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Extreme Downside Hedge Model

#### 2.2. Network (Graphical) Model

#### 2.3. Bayesian Estimation of Downside Risk Networks

- Sample via a Metropolis-within-Gibbs $\left[G\right|Y]$;
- Sample from $[B,{\mathrm{\Sigma}}_{\epsilon}|Y,G]$ by iterating the following steps:
- (a)
- Sample $\left[{B}_{i,{\pi}_{i}}\right|Y,G,{\mathrm{\Sigma}}_{\epsilon}]\sim \mathcal{N}({\widehat{B}}_{i,{\pi}_{i}},\phantom{\rule{3.33333pt}{0ex}}{D}_{{\pi}_{i}})$ where:$$\begin{array}{ccc}\hfill {\widehat{B}}_{i,{\pi}_{i}}={\sigma}_{\epsilon ,i}^{-2}{D}_{{\pi}_{i}}{X}_{{\pi}_{i}}^{\prime}{Y}_{i},& \phantom{\rule{1.em}{0ex}}\hfill & \hfill {D}_{{\pi}_{i}}={({\eta}^{-1}{I}_{{d}_{x}}+{\sigma}_{\epsilon ,i}^{-2}{X}_{{\pi}_{i}}^{\prime}{X}_{{\pi}_{i}})}^{-1}\end{array}$$
- (b)
- Sample $\left[{\mathrm{\Sigma}}_{\epsilon}^{-1}\right|Y,G,B]\sim \mathcal{W}(\delta +N,\phantom{\rule{3.33333pt}{0ex}}{\mathrm{\Lambda}}_{N})$ where:$$\begin{array}{c}\hfill {\mathrm{\Lambda}}_{N}={\mathrm{\Lambda}}_{0}+{(Y-X{\widehat{B}}^{\prime})}^{\prime}(Y-X{\widehat{B}}^{\prime})\end{array}$$

## 3. Data Description

## 4. Empirical Findings

#### Global Financial Crisis vs. COVID-19 Outbreak

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Billio, M.; Getmansky, M.; Lo, A.W.; Pelizzon, L. Econometric Measures of Connectedness and Systemic Risk in the Finance and Insurance Sectors. J. Financ. Econ.
**2012**, 104, 535–559. [Google Scholar] [CrossRef] - Diebold, F.; Yilmaz, K. On the Network Topology of Variance Decompositions: Measuring the Connectedness of Financial Firms. J. Econom.
**2014**, 182, 119–134. [Google Scholar] [CrossRef][Green Version] - Battiston, S.; Delli Gatti, D.; Gallegati, M.; Greenwald, B.; Stiglitz, J.E. Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk. J. Econ. Dyn. Control
**2012**, 36, 1121–1141. [Google Scholar] [CrossRef][Green Version] - Billio, M.; Casarin, R.; Rossini, L. Bayesian Nonparametric Sparse VAR Models. J. Econom.
**2019**, 212, 97–115. [Google Scholar] [CrossRef][Green Version] - Harris, R.D.; Nguyen, L.H.; Stoja, E. Systematic Extreme Downside Risk. J. Int. Financ. Mark. Inst. Money
**2019**, 61, 128–142. [Google Scholar] [CrossRef] - Ahelegbey, D.F.; Giudici, P.; Mojtahedi, F. Tail Risk Measurement In Crypto-Asset Markets. Int. Rev. Financ. Anal.
**2021**, 73, 101604. [Google Scholar] [CrossRef] - Mojtahedi, F.; Mojaverian, S.M.; Ahelegbey, D.F.; Giudici, P. Tail Risk Transmission: A Study of the Iran Food Industry. Risks
**2020**, 8, 78. [Google Scholar] [CrossRef] - Ahelegbey, D.F.; Billio, M.; Casarin, R. Bayesian Graphical Models for Structural Vector Autoregressive Processes. J. Appl. Econom.
**2016**, 31, 357–386. [Google Scholar] [CrossRef][Green Version] - Ahelegbey, D.F.; Billio, M.; Casarin, R. Sparse Graphical Vector Autoregression: A Bayesian Approach. Ann. Econ. Stat.
**2016**, 123/124, 333–361. [Google Scholar] [CrossRef] - Paci, L.; Consonni, G. Structural Learning of Contemporaneous Dependencies in Graphical VAR Models. Comput. Stat. Data Anal.
**2020**, 144, 106880. [Google Scholar] [CrossRef] - Gruber, L.F.; West, M. Bayesian Forecasting and Scalable Multivariate Volatility Analysis Using Simultaneous Graphical Dynamic Models. Econom. Stat.
**2017**, 3, 3–22. [Google Scholar] - Chabi-Yo, F.; Ruenzi, S.; Weigert, F. Crash Sensitivity and the Cross Section of Expected Stock Returns. J. Financ. Quant. Anal.
**2018**, 53, 1059–1100. [Google Scholar] [CrossRef] - Van Oordt, M.R.; Zhou, C. Systematic Tail Risk. J. Financ. Quant. Anal.
**2016**, 51, 685–705. [Google Scholar] [CrossRef] - Almeida, C.; Ardison, K.; Garcia, R.; Vicente, J. Nonparametric Tail Risk, Stock Returns and the Macroeconomy. J. Financ. Econom.
**2017**, 15, 333–376. [Google Scholar] - Hautsch, N.; Schaumburg, J.; Schienle, M. Financial Network Systemic Risk Contributions. Rev. Financ.
**2015**, 19, 685–738. [Google Scholar] [CrossRef][Green Version] - Härdle, W.K.; Wang, W.; Yu, L. TENET: Tail-Event driven NETwork risk. J. Econom.
**2016**, 192, 499–513. [Google Scholar] [CrossRef] - Wang, G.J.; Yi, S.; Xie, C.; Stanley, H.E. Multilayer Information Spillover Networks: Measuring Interconnectedness of Financial Institutions. Quant. Financ.
**2021**, 21, 1–23. [Google Scholar] [CrossRef] - Adrian, T.; Brunnermeier, M.K. CoVaR. Am. Econ. Rev.
**2016**, 106, 1705–1741. [Google Scholar] [CrossRef] - Geiger, D.; Heckerman, D. Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions. Ann. Stat.
**2002**, 30, 1412–1440. [Google Scholar] [CrossRef] - Ahelegbey, D.F.; Giudici, P. NetVIX-A Network Volatility Index of Financial Markets. Phys. A Stat. Mech. Its Appl.
**2022**, 594, 127017. [Google Scholar] [CrossRef] - Gelman, A.; Rubin, D.B. Inference from Iterative Simulation Using Multiple Sequences, (with discussion). Stat. Sci.
**1992**, 7, 457–511. [Google Scholar] [CrossRef] - Blume, L.; Easley, D.; Kleinberg, J.; Kleinberg, R.; Tardos, É. Network Formation in the Presence of Contagious Risk. ACM Trans. Econ. Comput.
**2013**, 1, 6. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Time series of daily equity log prices from January 1998 to December 2021, by regional classification: Americas (

**top**), Asia-Pacific (

**middle**), and Europe (

**bottom**).

**Figure 3.**Correlation of $\Delta {\mathrm{Net}\text{-}\mathrm{Density}}_{t+h}$ with $\Delta {\mathrm{VIX}}_{t}$.

**Figure 4.**Sub-period downside risk transmission network among stock markets. Red nodes represent the markets of the Americas, blue for Europe, and green for Asia-Pacific. The size of the nodes is out-degree weighted. Red links denote negative sensitivity and green for positive reactions.

**Figure 5.**Comparing the global financial crisis (GFC) and COVID-19 outbreak network. Red nodes represent the markets of the Americas, blue for Europe, and green for Asia-Pacific. The size of the nodes is out-degree weighted. Red links denote negative effects and green for positive interactions. The number in parenthesis signifies the total links in each network. Note: ${A}^{c}$—complement of A.

**Table 1.**Detailed description of the stock market indices of countries classified according to regions.

Region | No. | Country | Code | Description | Index |
---|---|---|---|---|---|

1 | Argentina | AR | Argentina MERVAL | MERVAL | |

2 | Brazil | BR | Brazil Bovespa | IBOV | |

Americas | 3 | Canada | CA | Canada TSX Comp. | SPTSX |

4 | Mexico | MX | Mexico IPC | MEXBOL | |

5 | United States | US | United States S&P 500 | SPX | |

6 | Australia | AU | Australia ASX 200 | AS51 | |

7 | China | CN | China SSE Comp. | SHCOMP | |

Asia-Pacific | 8 | Hong Kong | HK | Hong Kong Hang Seng | HSI |

9 | India | IN | India BSE Sensex | SENSEX | |

10 | Japan | JP | Japan Nikkei 225 | NKY | |

11 | Korea | KR | South Korean KOSPI | KOSPI | |

12 | Belgium | BE | Belgium BEL 20 | BEL20 | |

13 | France | FR | France CAC 40 | CAC | |

14 | Germany | DE | Germany DAX 30 | DAX | |

15 | Italy | IT | Italy FTSE MIB | FTSEMIB | |

Europe | 16 | The Netherlands | NL | The Netherlands AEX | AEX |

17 | Russia | RU | Russia MOEX | IMOEX | |

18 | Spain | ES | Spain IBEX 35 | IBEX | |

19 | Switzerland | CH | Switzerland SMI | SMI | |

20 | United Kingdom | UK | U.K. FTSE 100 | UKX |

**Table 2.**Statistics of daily returns and change in the ES for stock markets (March 1998–December 2020).

Daily Returns | Daily Log ES | |||||||
---|---|---|---|---|---|---|---|---|

Code | Mean | SD | Skew | Kurt | Mean | SD | Skew | Kurt |

AR | 0.0775 | 2.2776 | −1.7342 | 35.0728 | −0.0154 | 63.4156 | −5.9109 | 820.8923 |

BR | 0.0373 | 1.9175 | 0.2348 | 15.7679 | −0.0249 | 64.1751 | −21.0317 | 1841.0830 |

CA | 0.0182 | 1.0978 | −0.9401 | 16.8512 | −0.0298 | 21.5994 | 1.8971 | 254.6383 |

MX | 0.0384 | 1.3268 | 0.1088 | 6.4837 | −0.0678 | 22.4015 | 0.3937 | 74.0335 |

US | 0.0246 | 1.2155 | −0.3997 | 10.9808 | −0.0285 | 20.8834 | −1.4264 | 133.3279 |

AU | 0.0168 | 0.9926 | −0.6961 | 8.5250 | −0.0150 | 16.3888 | 1.4337 | 138.1182 |

CN | 0.0171 | 1.4756 | −0.3506 | 5.6949 | −0.0557 | 38.1862 | −2.6821 | 189.0206 |

HK | 0.0136 | 1.4693 | −0.0185 | 7.1981 | −0.0925 | 25.9796 | −0.8570 | 142.6127 |

IN | 0.0463 | 1.4532 | −0.2802 | 8.8733 | −0.0176 | 43.0280 | −5.6080 | 812.9438 |

JP | 0.0087 | 1.4343 | −0.3390 | 6.4696 | −0.0055 | 31.0559 | −3.3666 | 219.1089 |

KR | 0.0284 | 1.5794 | −0.2769 | 6.4429 | −0.0696 | 29.0712 | −0.1575 | 269.9062 |

BE | 0.0075 | 1.2404 | −0.4001 | 9.5224 | −0.0072 | 22.4958 | 1.6991 | 239.2680 |

FR | 0.0128 | 1.4137 | −0.2117 | 6.2409 | −0.0191 | 26.2744 | −3.4356 | 280.8779 |

DE | 0.0202 | 1.4631 | −0.1968 | 5.6709 | −0.0334 | 23.6652 | 0.0171 | 143.6197 |

IT | −0.0008 | 1.5230 | −0.5382 | 8.5133 | −0.0058 | 31.5024 | −5.9900 | 457.8811 |

NL | 0.0096 | 1.3822 | −0.2423 | 7.0535 | −0.0097 | 21.5194 | 0.1786 | 106.0908 |

RU | 0.0648 | 2.2908 | 0.2812 | 19.9011 | −0.1418 | 45.9988 | −0.4843 | 141.6174 |

ES | 0.0014 | 1.4612 | −0.3061 | 7.6552 | −0.0023 | 33.7401 | 0.4265 | 293.9172 |

CH | 0.0095 | 1.1575 | −0.2895 | 7.5521 | −0.0252 | 21.6889 | 3.1533 | 219.5005 |

UK | 0.0044 | 1.1715 | −0.3089 | 7.5820 | −0.0173 | 18.6884 | −0.1894 | 96.3411 |

No. | Sub-Period | Average Degree | Density | Clustering Coefficient | Average Path Length |
---|---|---|---|---|---|

1 | Pre-GFC | 1.150 | 6.053 | 0.733 | 1.324 |

2 | GFC | 6.550 | 34.474 | 0.754 | 2.211 |

3 | Pre-COVID-19 | 1.100 | 5.789 | 0.814 | 1.083 |

4 | COVID-19 | 6.900 | 36.316 | 0.787 | 1.734 |

Rank | GFC | COVID | ||||||
---|---|---|---|---|---|---|---|---|

Hub | Auth | Hub | Auth | |||||

1 | DE | 0.471 | FR | 0.884 | CH | 0.411 | NL | 0.560 |

2 | IT | 0.378 | NL | 0.252 | RU | 0.335 | FR | 0.454 |

3 | NL | 0.373 | UK | 0.211 | ES | 0.330 | ES | 0.329 |

4 | UK | 0.367 | IT | 0.158 | DE | 0.308 | DE | 0.314 |

5 | ES | 0.354 | ES | 0.141 | BR | 0.294 | CA | 0.269 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ahelegbey, D.F.
Statistical Modelling of Downside Risk Spillovers. *FinTech* **2022**, *1*, 125-134.
https://doi.org/10.3390/fintech1020009

**AMA Style**

Ahelegbey DF.
Statistical Modelling of Downside Risk Spillovers. *FinTech*. 2022; 1(2):125-134.
https://doi.org/10.3390/fintech1020009

**Chicago/Turabian Style**

Ahelegbey, Daniel Felix.
2022. "Statistical Modelling of Downside Risk Spillovers" *FinTech* 1, no. 2: 125-134.
https://doi.org/10.3390/fintech1020009