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Non-parametric determination of real-time lag structure between two time series: the 'optimal thermal causal path' method

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  • Didier Sornette
  • Wei-Xing Zhou

Abstract

We introduce a novel non-parametric methodology to test for the dynamical time evolution of the lag-lead structure between two arbitrary time series. The method consists of constructing a distance matrix based on the matching of all sample data pairs between the two time series. Then, the lag-lead structure is searched for as the optimal path in the distance matrix landscape that minimizes the total mismatch between the two time series, and that obeys a one-to-one causal matching condition. To make the solution robust to the presence of a large amount of noise that may lead to spurious structures in the distance matrix landscape, we generalize this optimal search by introducing a fuzzy search by sampling over all possible paths, each path being weighted according to a multinomial logit or equivalently Boltzmann factor proportional to the exponential of the global mismatch of this path. We present the efficient transfer matrix method that solves the problem and test it on simple synthetic examples to demonstrate its properties and usefulness compared with the standard running-time cross-correlation method. We then apply our 'optimal thermal causal path' method to the question of the lag-dependence between the US stock market and the treasury bond yields and confirm our earlier results on an arrow of the stock markets preceding the Federal Reserve Funds' adjustments, as well as the yield rates at short maturities in the period 2000-2003. Our application of this technique to inflation, inflation change, GDP growth rate and unemployment rate unearths non-trivial lag relationships: the GDP changes lead inflation especially since the 1980s, inflation changes leads GDP only in the 1980 decade, and inflation leads unemployment rates since the 1970s. In addition, our approach seems to detect multiple competing lag structures in which one can have inflation leading GDP with a certain lag time and GDP feeding back/leading inflation with another lag time.

Suggested Citation

  • Didier Sornette & Wei-Xing Zhou, 2005. "Non-parametric determination of real-time lag structure between two time series: the 'optimal thermal causal path' method," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 577-591.
  • Handle: RePEc:taf:quantf:v:5:y:2005:i:6:p:577-591
    DOI: 10.1080/14697680500383763
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Zhou, Wei-Xing & Sornette, Didier, 2007. "Lead-lag cross-sectional structure and detection of correlated–anticorrelated regime shifts: Application to the volatilities of inflation and economic growth rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 287-296.
    2. Yao, Can-Zhong & Lin, Ji-Nan & Lin, Qing-Wen & Zheng, Xu-Zhou & Liu, Xiao-Feng, 2016. "A study of causality structure and dynamics in industrial electricity consumption based on Granger network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 297-320.
    3. repec:taf:quantf:v:17:y:2017:i:6:p:959-977 is not listed on IDEAS
    4. repec:eee:phsmap:v:483:y:2017:i:c:p:299-308 is not listed on IDEAS
    5. repec:eee:phsmap:v:486:y:2017:i:c:p:535-541 is not listed on IDEAS
    6. Xu, Hai-Chuan & Zhou, Wei-Xing & Sornette, Didier, 2017. "Time-dependent lead-lag relationship between the onshore and offshore Renminbi exchange rates," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 49(C), pages 173-183.
    7. repec:eee:phsmap:v:523:y:2019:i:c:p:723-733 is not listed on IDEAS
    8. Jia, Rui-Lin & Wang, Dong-Hua & Tu, Jing-Qing & Li, Sai-Ping, 2016. "Correlation between agricultural markets in dynamic perspective—Evidence from China and the US futures markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 464(C), pages 83-92.
    9. Kun GUO & Wei-Xing ZHOU & Si-Wei CHENG & Didier SORNETTE, 2011. "The US stock market leads the Federal funds rate and Treasury bond yields," Swiss Finance Institute Research Paper Series 11-05, Swiss Finance Institute.
    10. Paul Gaskell & Frank McGroarty & Thanassis Tiropanis, 2014. "Signal Diffusion Mapping: Optimal Forecasting with Time Varying Lags," Papers 1409.6443, arXiv.org.
    11. Stübinger, Johannes, 2018. "Statistical arbitrage with optimal causal paths on high-frequencydata of the S&P 500," FAU Discussion Papers in Economics 01/2018, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    12. Guo, Kun & Sun, Yi & Qian, Xin, 2017. "Can investor sentiment be used to predict the stock price? Dynamic analysis based on China stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 390-396.
    13. Wang, Xuan & Guo, Kun & Lu, Xiaolin, 2016. "The long-run dynamic relationship between exchange rate and its attention index: Based on DCCA and TOP method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 108-115.
    14. Hao Meng & Hai-Chuan Xu & Wei-Xing Zhou & Didier Sornette, 2017. "Symmetric thermal optimal path and time-dependent lead-lag relationship: novel statistical tests and application to UK and US real-estate and monetary policies," Quantitative Finance, Taylor & Francis Journals, vol. 17(6), pages 959-977, June.
    15. repec:eee:phsmap:v:513:y:2019:i:c:p:709-723 is not listed on IDEAS
    16. Gong, Chen-Chen & Ji, Shen-Dan & Su, Li-Ling & Li, Sai-Ping & Ren, Fei, 2016. "The lead–lag relationship between stock index and stock index futures: A thermal optimal path method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 63-72.

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