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Estimation of the lead–lag parameter between two stochastic processes driven by fractional Brownian motions

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  • Kohei Chiba

    (The University of Tokyo)

Abstract

In this paper, we consider the problem of estimating the lead–lag parameter between two stochastic processes driven by fractional Brownian motions (fBMs) of the Hurst parameter greater than 1/2. First we propose a lead–lag model between two stochastic processes involving fBMs, and then construct a consistent estimator of the lead–lag parameter with possible convergence rate. Our estimator has the following two features. Firstly, we can construct the lead–lag estimator without using the Hurst parameters of the underlying fBMs. Secondly, our estimator can deal with some non-synchronous and irregular observations. We explicitly calculate possible convergence rate when the observation times are (1) synchronous and equidistant, and (2) given by the Poisson sampling scheme. We also present numerical simulations of our results using the R package YUIMA.

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  • 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.
    • Handle: RePEc:spr:sistpr:v:22:y:2019:i:3:d:10.1007_s11203-018-09195-5
    DOI: 10.1007/s11203-018-09195-5
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    References listed on IDEAS

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    1. Takaki Hayashi & Nakahiro Yoshida, 2008. "Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 367-406, June.
    2. 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.
    3. Takaki Hayashi & Shigeo Kusuoka, 2008. "Consistent estimation of covariation under nonsynchronicity," Statistical Inference for Stochastic Processes, Springer, vol. 11(1), pages 93-106, February.
    4. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
    5. Koike, Yuta, 2014. "Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2699-2753.
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    Cited by:

    1. Hayashi, Takaki & Koike, Yuta, 2019. "No arbitrage and lead–lag relationships," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.

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