Direct Estimation of Lead–Lag Relationships Using Multinomial Dynamic Time Warping


This paper investigates the lead–lag relationships in high-frequency data. We propose multinomial dynamic time warping (MDTW) that deals with non-synchronous observation, vast data, and time-varying lead–lag. MDTW directly estimates the lead–lags without lag candidates. Its computational complexity is linear with respect to the number of observation and it does not depend on the number of lag candidates. The experiments adopting artificial data and market data illustrate the effectiveness of our method compared to the existing methods.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.


  1. Brock, W. A., & Kleidon, A. W. (1992). Periodic market closure and trading volume. Journal of Economic Dynamics and Control, 16(3–4), 451–489.

    Article  Google Scholar 

  2. Chiba, K. (2017). Estimation of the lead-lag parameter between two stochastic processes driven by fractional Brownian motions. arXiv:1705.10466.

  3. Dobreva, D., & Schaumburgb, E. (2017). High-frequency cross-market trading: Model free measurement and applications. Working paper.

  4. Eun, C. S., & Shim, S. (1989). International transmission of stock market movements. The Journal of Financial and Quantitative Analysis, 24(2), 241.

    Article  Google Scholar 

  5. Garbade, K. D., & Silber, W. L. (1979). Dominant and satellite markets: A study of dually-traded securities. The Review of Economics and Statistics, 61(3), 455.

    Article  Google Scholar 

  6. Hayashi, T., & Koike, Y. (2017). Multi-scale analysis of lead-lag relationships in high-frequency financial markets. arXiv:1708.03992.

  7. Hayashi, T., & Koike, Y. (2018). Wavelet-based methods for high-frequency lead-lag analysis. SIAM Journal on Financial Mathematics, 9(4), 1208–1248.

    Article  Google Scholar 

  8. Hayashi, T., & Yoshida, N. (2005). On covariance estimation of non-synchronously observed diffusion processes. Bernoulli, 11(2), 359–379.

    Article  Google Scholar 

  9. Hendershott, T., & Riordan, R. (2012). High frequency trading and price discovery. SSRN Electronic Journal.

    Article  Google Scholar 

  10. Hoffmann, M., Rosenbaum, M., & Yoshida, N. (2013). Estimation of the lead-lag parameter from non-synchronous data. Bernoulli, 19(2), 426–461.

    Article  Google Scholar 

  11. Huth, N., & Abergel, F. (2014). High frequency lead/lag relationships: Empirical facts. Journal of Empirical Finance, 26, 41–58.

    Article  Google Scholar 

  12. Ito, T., Lyons, R. K., & Melvin, M. T. (1998). Is there private information in the FX market? The Tokyo experiment. The Journal of Finance, 53(3), 1111–1130.

    Article  Google Scholar 

  13. Kawaller, I. G., Koch, P. D., & Koch, T. W. (1987). The temporal price relationship between s&p 500 futures and the s&p 500 index. The Journal of Finance, 42(5), 1309–1329.

    Article  Google Scholar 

  14. Lo, A. W., & MacKinlay, A. C. (1990). An econometric analysis of nonsynchronous trading. Journal of Econometrics, 45(1–2), 181–211.

    Article  Google Scholar 

  15. Lucca, D. O., & Moench, E. (2015). The pre-FOMC announcement drift. The Journal of Finance, 70(1), 329–371.

    Article  Google Scholar 

  16. Müller, M. (2007). Information retrieval for music and motion. Berlin: Springer.

    Book  Google Scholar 

  17. Peiers, B. (1997). Informed traders, intervention, and price leadership: A deeper view of the microstructure of the foreign exchange market. The Journal of Finance, 52(4), 1589–1614.

    Article  Google Scholar 

  18. Raihan, T. (2017). Predicting US recessions: A dynamic time warping exercise in economics. SSRN Electronic Journal,.

    Article  Google Scholar 

  19. Salvador, S., & Chan, P. (2007). FastDTW: Toward accurate dynamic time warping in linear time and space. Intelligent Data Analysis, 11(5), 561–580.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Katsuya Ito.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ito, K., Sakemoto, R. Direct Estimation of Lead–Lag Relationships Using Multinomial Dynamic Time Warping. Asia-Pac Financ Markets 27, 325–342 (2020).

Download citation


  • Lead–lag relationships
  • High frequency trading
  • Dynamic time warping

JEL Classification

  • C63
  • C58