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On the limiting spectral distribution of the covariance matrices of time-lagged processes

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  • Robert, Christian Y.
  • Rosenbaum, Mathieu

Abstract

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we provide an asymptotic theory for our estimation procedure.

Suggested Citation

  • Robert, Christian Y. & Rosenbaum, Mathieu, 2010. "On the limiting spectral distribution of the covariance matrices of time-lagged processes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2434-2451, November.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:10:p:2434-2451
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    1. de Jong, Frank & Nijman, Theo, 1997. "High frequency analysis of lead-lag relationships between financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 259-277, June.
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    3. Kawaller, Ira G & Koch, Paul D & Koch, Timothy W, 1987. "The Temporal Price Relationship between S&P 500 Futures and the S and P 500 Index," Journal of Finance, American Finance Association, vol. 42(5), pages 1309-1329, December.
    4. Yin, Y. Q. & Krishnaiah, P. R., 1983. "A limit theorem for the eigenvalues of product of two random matrices," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 489-507, December.
    5. Nijman, T.E. & de Jong, F.C.J.M., 1997. "High frequency analysis of lead-lag relationships between financial markets," Other publications TiSEM f4f406a0-771a-4af2-9364-6, Tilburg University, School of Economics and Management.
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    Cited by:

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    2. Hayashi, Takaki & Koike, Yuta, 2019. "No arbitrage and lead–lag relationships," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.
    3. Kartikay Gupta & Niladri Chatterjee, 2020. "Examining Lead-Lag Relationships In-Depth, With Focus On FX Market As Covid-19 Crises Unfolds," Papers 2004.10560, arXiv.org, revised May 2020.
    4. Takaki Hayashi & Yuta Koike, 2017. "No arbitrage and lead-lag relationships," Papers 1712.09854, arXiv.org.
    5. Takaki Hayashi & Yuta Koike, 2016. "Wavelet-based methods for high-frequency lead-lag analysis," Papers 1612.01232, arXiv.org, revised Nov 2018.
    6. Kohei Chiba, 2019. "Estimation of the lead–lag parameter between two stochastic processes driven by fractional Brownian motions," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 323-357, October.

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