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Sparse inference of the drift of a high-dimensional Ornstein–Uhlenbeck process

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  • Gaïffas, Stéphane
  • Matulewicz, Gustaw

Abstract

Given the observation of a high-dimensional Ornstein–Uhlenbeck (OU) process in continuous time, we are interested in inference on the drift parameter under a row-sparsity assumption. Towards that aim, we consider the negative log-likelihood of the process, penalized by an ℓ1-penalization (Lasso and Adaptive Lasso). We provide both finite- and large-sample results for this procedure, by means of a sharp oracle inequality, and a limit theorem in the long-time asymptotics, including asymptotic consistency for variable selection. As a by-product, we point out the fact that for the Ornstein–Uhlenbeck process, one does not need an assumption of restricted eigenvalue type in order to derive fast rates for the Lasso, while it is well-known to be mandatory for linear regression for instance. Numerical results illustrate the benefits of this penalized procedure compared to standard maximum likelihood approaches both on simulations and real-world financial data.

Suggested Citation

  • Gaïffas, Stéphane & Matulewicz, Gustaw, 2019. "Sparse inference of the drift of a high-dimensional Ornstein–Uhlenbeck process," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 1-20.
    • Handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:1-20
    DOI: 10.1016/j.jmva.2018.08.005
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Blasques, Francisco & Bräuning, Falk & Lelyveld, Iman van, 2018. "A dynamic network model of the unsecured interbank lending market," Journal of Economic Dynamics and Control, Elsevier, vol. 90(C), pages 310-342.
    3. Gabrieli, Silvia & Georg, Co-Pierre, 2014. "A network view on interbank market freezes," Discussion Papers 44/2014, Deutsche Bundesbank.
    4. Blasques, Francisco & Bräuning, Falk & Lelyveld, Iman van, 2018. "A dynamic network model of the unsecured interbank lending market," Journal of Economic Dynamics and Control, Elsevier, vol. 90(C), pages 310-342.
    5. van Zanten, Harry, 2000. "A multivariate central limit theorem for continuous local martingales," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 229-235, November.
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    Cited by:

    1. Alessandro Gregorio & Francesco Iafrate, 2021. "Regularized bridge-type estimation with multiple penalties," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 921-951, October.
    2. Junichiro Yoshida & Nakahiro Yoshida, 2024. "Quasi-maximum likelihood estimation and penalized estimation under non-standard conditions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(5), pages 711-763, October.
    3. Valentin Courgeau & Almut E. D. Veraart, 2022. "Likelihood theory for the graph Ornstein-Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 227-260, July.
    4. Junichiro Yoshida & Nakahiro Yoshida, 2024. "Penalized estimation for non-identifiable models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(5), pages 765-796, October.

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