This paper presents a method to identify causal interactions between two time series. The largest eigenvalue follows a Tracy-Widom distribution, derived from a Coulomb gas model. This defines causal interactions as the pushing and pulling of the gas, measurable by the variability of the largest eigenvalue's explanatory power. The hypothesis that this setup applies to time series interactions was validated, with causality inferred from time lags. The standard deviation of the largest eigenvalue's explanatory power in lagged correlation matrices indicated the probability of causal interaction between time series. Contrasting with traditional methods that rely on forecasting or window-based parametric controls, this approach offers a novel definition of causality based on dynamic monitoring of tail events. Experimental validation with controlled trials and historical data shows that this method outperforms Granger's causality test in detecting structural changes in time series. Applications to stock returns and financial market data show the indicator's predictive capabilities regarding average stock return and realized volatility. Further validation with brokerage data confirms its effectiveness in inferring causal relationships in liquidity flows, highlighting its potential for market and liquidity risk management.
Citation: Alejandro Rodriguez Dominguez, Om Hari Yadav. A causal interactions indicator between two time series using extreme variations in the first eigenvalue of lagged correlation matrices[J]. Data Science in Finance and Economics, 2024, 4(3): 422-445. doi: 10.3934/DSFE.2024018
This paper presents a method to identify causal interactions between two time series. The largest eigenvalue follows a Tracy-Widom distribution, derived from a Coulomb gas model. This defines causal interactions as the pushing and pulling of the gas, measurable by the variability of the largest eigenvalue's explanatory power. The hypothesis that this setup applies to time series interactions was validated, with causality inferred from time lags. The standard deviation of the largest eigenvalue's explanatory power in lagged correlation matrices indicated the probability of causal interaction between time series. Contrasting with traditional methods that rely on forecasting or window-based parametric controls, this approach offers a novel definition of causality based on dynamic monitoring of tail events. Experimental validation with controlled trials and historical data shows that this method outperforms Granger's causality test in detecting structural changes in time series. Applications to stock returns and financial market data show the indicator's predictive capabilities regarding average stock return and realized volatility. Further validation with brokerage data confirms its effectiveness in inferring causal relationships in liquidity flows, highlighting its potential for market and liquidity risk management.
| [1] |
Arkol O, Azimli A (2024) Pricing the common stocks in emerging markets: The role of economic policy uncertainty. Modern Financ 2: 31–50. https://doi.org/10.61351/mf.v2i1.93 doi: 10.61351/mf.v2i1.93
|
| [2] |
Balcilar M, Gungor H, Hammoudeh H (2015) The time-varying causality between spot and futures crude oil prices: A regime switching approach. Int Rev Econ Financ 40: 51–71. https://doi.org/10.1016/j.iref.2015.02.008. doi: 10.1016/j.iref.2015.02.008
|
| [3] | Bouchaud JP, Potters M (2009) Financial Applications of Random Matrix Theory: a short review. arXiv.org. Quantitative Finance Papers. |
| [4] |
Breitung J, Candelon B (2006) Testing for Short and Long-Run Causality: A Frequency Domain Approach. J Econometrics 132: 363–378. https://doi.org/10.1016/j.jeconom.2005.02.004. doi: 10.1016/j.jeconom.2005.02.004
|
| [5] | Brown P, Walsh D, Yuen A (1997) The interaction between order imbalance and stock price. Pac-Basin Financ J 5: 539–557. |
| [6] |
Brukner Č (2014) Quantum causality. Nature Phys 10: 259–263. https://doi.org/10.1038/nphys2930. doi: 10.1038/nphys2930
|
| [7] | Cochrane JH (1997) Time Series for Macroeconomics and Finance. Graduate School of Business, University of Chicago, Chicago. |
| [8] |
Cunden FD, Facchi P, Ligabò M, et al. (2018) Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls. J Stat Phys 175: 1262–1297. https://doi.org/10.1007/s10955-019-02281-9 doi: 10.1007/s10955-019-02281-9
|
| [9] |
Edinburgh T, Eglen SJ, Ercole A (2021) Causality indices for bivariate time series data: A comparative review of performance. Chaos 31: 083111. https://doi.org/10.1063/5.0053519 doi: 10.1063/5.0053519
|
| [10] |
Fenghua W, Jihong X, Chuangxia H, et al. (2017) Interaction between oil and US dollar exchange rate: nonlinear causality, time-varying influence and structural breaks in volatility. Appl Econ 50: 1–16. https://doi.org/10.1080/00036846.2017.1321838. doi: 10.1080/00036846.2017.1321838
|
| [11] |
Granger CWJ (1969) Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica 37: 424–438. https://doi.org/10.2307/1912791 doi: 10.2307/1912791
|
| [12] |
Hastings SP, McLeod JB (1980) A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation. Arch Rational Mech Anal 73: 31–51. https://doi.org/10.1007/BF00283254 doi: 10.1007/BF00283254
|
| [13] | Havas P (1968) Causality Requirements and the Theory of Relativity. Synthese 18: 75–102. Available from: http://www.jstor.org/stable/20114593. |
| [14] |
Janse R, Hoekstra T, Jager K, et al. (2021) Conducting correlation analysis: important limitations and pitfalls. Clin Kidney J 14: 2332–2337. https://doi.org/10.1093/ckj/sfab085 doi: 10.1093/ckj/sfab085
|
| [15] | Karimi K, Hamilton HJ (2011) Generation and Interpretation of Temporal Decision Rules. Int J Comput Inf Syst Ind Manag Appl 3: 314–323. |
| [16] |
Kosuke I, Keele L, Yamamoto T (2010) Identification, Inference and Sensitivity Analysis for Causal Mediation Effects. Stat Sci 25: 51–71. https://doi.org/10.1214/10-STS321. doi: 10.1214/10-STS321
|
| [17] |
Li J, Wu B, Sun X, et al. (2021) Causal Hidden Markov Model for Time Series Disease Forecasting. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 12100–12109. https://doi.org/10.1109/CVPR46437.2021.01193 doi: 10.1109/CVPR46437.2021.01193
|
| [18] |
Majumdar SN, Schehr G (2014) Top eigenvalue of a random matrix: Large deviations and third order phase transition. J Stat Mech-Theory E. https://doi.org/10.1088/1742-5468/2014/01/P01012 doi: 10.1088/1742-5468/2014/01/P01012
|
| [19] | Marti G, Nielsen F, Bińkowski M, et al. (2021) A Review of Two Decades of Correlations, Hierarchies, Networks and Clustering in Financial Markets. arXiv. |
| [20] |
Mensi W, Maitra D, Vinh VX, et al. (2021) Asymmetric volatility connectedness among main international stock markets: A high frequency analysis. Borsa Istanb Rev 21: 291–306. https://doi.org/10.1016/j.bir.2020.12.003 doi: 10.1016/j.bir.2020.12.003
|
| [21] |
Nadal C, Majumdar SN (2011) A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix. J Stat Mech-Theory Exp. https://doi.org/10.1088/1742-5468/2011/04/P04001. doi: 10.1088/1742-5468/2011/04/P04001
|
| [22] | Nydick SW (2012) The Wishart and Inverse Wishart Distributions. Electron J Stat 6: 1–19. |
| [23] | Pearl J (2009) Causality: Models, Reasoning and Inference (2nd. ed.). Cambridge University Press, USA. |
| [24] | Potters M, Bouchaud JP, Laloux L (2005) Financial Applications of Random Matrix Theory: Old Laces and New Pieces. Science & Finance. Capital Fund Management, Science & Finance (CFM) working paper archive, 36. |
| [25] | Rodriguez Dominguez A, Stynes D (2022) A Clustering Algorithm for Correlation Quickest Hub Discovery Mixing Time Evolution and Random Matrix Theory. 2022 IEEE 34th International Conference on Tools with Artificial Intelligence (ICTAI), 1007–1014. |
| [26] |
Rosenfelder R (1989) Causality and the Coulomb sum rule in nuclei. Phys Rev C 39: 2166–2169. https://doi.org/10.1103/PhysRevC.39.2166 doi: 10.1103/PhysRevC.39.2166
|
| [27] |
Majumdar S., Pal A. and Schehr G. (2020). Extreme value statistics of correlated random variables: A pedagogical review. Phys Rep 840: 1–32. https://doi.org/10.1016/j.physrep.2019.10.005 doi: 10.1016/j.physrep.2019.10.005
|
| [28] |
Samsul A, Shahzad SJH, Román F (2019) Causal flows between oil and forex markets using high-frequency data: Asymmetries from good and bad volatility. Energ Econ 84: 104513. https://doi.org/10.1016/j.eneco.2019.104513 doi: 10.1016/j.eneco.2019.104513
|
| [29] | Sklar L (2009) Causation in Statistical Mechanics. In Helen Beebee, Christopher Hitchcock & Peter Menzies (eds.), The Oxford Handbook of Causation. Oxford University Press. |
| [30] | Tracy CA, Widom H (2002) Distribution functions for largest eigenvalues and their applications. |
| [31] | Yarlagadda R, Hershey J (2003) Signal Processing, General, In: Editor(s): Robert A. Meyers, Encyclopedia of Physical Science and Technology (Third Edition), Academic Press, 761–779. |
Figures(19) / Tables(3)
Alejandro Rodriguez Dominguez, Om Hari Yadav. A causal interactions indicator between two time series using extreme variations in the first eigenvalue of lagged correlation matrices[J]. Data Science in Finance and Economics, 2024, 4(3): 422-445. doi: 10.3934/DSFE.2024018
DownLoad: