IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v25y2022i2d10.1007_s11203-021-09257-1.html
   My bibliography  Save this article

Likelihood theory for the graph Ornstein-Uhlenbeck process

Author

Listed:
  • Valentin Courgeau

    (Imperial College London)

  • Almut E. D. Veraart

    (Imperial College London)

Abstract

We consider the problem of modelling restricted interactions between continuously-observed time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph Ornstein-Uhlenbeck (GrOU) process driven by a general Lévy process to study the momentum and network effects amongst nodes, effects that quantify the impact of a node on itself and that of its neighbours, respectively. We derive the maximum likelihood estimators (MLEs) and their usual properties (existence, uniqueness and efficiency) along with their asymptotic normality and consistency. Additionally, an Adaptive Lasso approach, or a penalised likelihood scheme, infers both the graph structure along with the GrOU parameters concurrently and is shown to satisfy similar properties. Finally, we show that the asymptotic theory extends to the case when stochastic volatility modulation of the driving Lévy process is considered.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sistpr:v:25:y:2022:i:2:d:10.1007_s11203-021-09257-1
    DOI: 10.1007/s11203-021-09257-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11203-021-09257-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11203-021-09257-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kevei, Péter, 2018. "Ergodic properties of generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 156-181.
    2. S. Endres & J. Stübinger, 2019. "Optimal trading strategies for Lévy-driven Ornstein–Uhlenbeck processes," Applied Economics, Taylor & Francis Journals, vol. 51(29), pages 3153-3169, June.
    3. Pap, Gyula & van Zuijlen, Martien C. A., 1996. "Parameter Estimation with Exact Distribution for Multidimensional Ornstein-Uhlenbeck Processes," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 153-165, November.
    4. Kim, Donggyu & Wang, Yazhen & Zou, Jian, 2016. "Asymptotic theory for large volatility matrix estimation based on high-frequency financial data," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3527-3577.
    5. Yamazato, Makoto, 1983. "Absolute continuity of operator-self-decomposable distributions on Rd," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 550-560, December.
    6. Cecilia Mancini, 2009. "Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 270-296, June.
    7. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    8. Zhu, Xuening & Huang, Danyang & Pan, Rui & Wang, Hansheng, 2020. "Multivariate spatial autoregressive model for large scale social networks," Journal of Econometrics, Elsevier, vol. 215(2), pages 591-606.
    9. Christian Brownlees & Eulalia Nualart & Yucheng Sun, 2020. "On the estimation of integrated volatility in the presence of jumps and microstructure noise," Econometric Reviews, Taylor & Francis Journals, vol. 39(10), pages 991-1013, November.
    10. 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.
    11. Sims, Christopher A, 1980. "Macroeconomics and Reality," Econometrica, Econometric Society, vol. 48(1), pages 1-48, January.
    12. 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.
    13. Chen, Cathy Yi-Hsuan & Härdle, Wolfgang Karl & Okhrin, Yarema, 2019. "Tail event driven networks of SIFIs," Journal of Econometrics, Elsevier, vol. 208(1), pages 282-298.
    14. Ole E. Barndorff-Nielsen & Almut E. D. Veraart, 2012. "Stochastic Volatility of Volatility and Variance Risk Premia," Journal of Financial Econometrics, Oxford University Press, vol. 11(1), pages 1-46, December.
    15. Valentin Courgeau & Almut E. D. Veraart, 2020. "High-frequency Estimation of the L\'evy-driven Graph Ornstein-Uhlenbeck process," Papers 2008.10930, arXiv.org, revised Jul 2022.
    16. Jan Rosiński & Tomasz Żak, 1997. "The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes," Journal of Theoretical Probability, Springer, vol. 10(1), pages 73-86, January.
    17. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
    18. Alexander Gushchin & Ilya Pavlyukevich & Marian Ritsch, 2020. "Drift estimation for a Lévy-driven Ornstein–Uhlenbeck process with heavy tails," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 553-570, October.
    19. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    20. Fasen, Vicky, 2013. "Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration," Journal of Econometrics, Elsevier, vol. 172(2), pages 325-337.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kevin W. Lu, 2022. "Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 365-396, July.
    2. A. Aghamohammadi & S. Mohammadi, 2017. "Bayesian analysis of penalized quantile regression for longitudinal data," Statistical Papers, Springer, vol. 58(4), pages 1035-1053, December.
    3. Todorov, Viktor & Tauchen, George & Grynkiv, Iaryna, 2014. "Volatility activity: Specification and estimation," Journal of Econometrics, Elsevier, vol. 178(P1), pages 180-193.
    4. Matteo Barigozzi & Marc Hallin, 2017. "A network analysis of the volatility of high dimensional financial series," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(3), pages 581-605, April.
    5. Dong, Ruipeng & Li, Daoji & Zheng, Zemin, 2021. "Parallel integrative learning for large-scale multi-response regression with incomplete outcomes," Computational Statistics & Data Analysis, Elsevier, vol. 160(C).
    6. Palandri, Alessandro, 2015. "Do negative and positive equity returns share the same volatility dynamics?," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 486-505.
    7. Behme, Anita & Di Tella, Paolo & Sideris, Apostolos, 2024. "On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    8. Bercu, Bernard & Proïa, Frédéric & Savy, Nicolas, 2014. "On Ornstein–Uhlenbeck driven by Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 36-44.
    9. Zachary F. Fisher & Younghoon Kim & Barbara L. Fredrickson & Vladas Pipiras, 2022. "Penalized Estimation and Forecasting of Multiple Subject Intensive Longitudinal Data," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 1-29, June.
    10. 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.
    11. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
    12. Yiying Cheng & Yaozhong Hu & Hongwei Long, 2020. "Generalized moment estimators for $$\alpha $$α-stable Ornstein–Uhlenbeck motions from discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 53-81, April.
    13. Li Guo & Wolfgang Karl Härdle & Yubo Tao, 2024. "A Time-Varying Network for Cryptocurrencies," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 42(2), pages 437-456, April.
    14. Laurent, Sébastien & Shi, Shuping, 2020. "Volatility estimation and jump detection for drift–diffusion processes," Journal of Econometrics, Elsevier, vol. 217(2), pages 259-290.
    15. Matteo Barigozzi & Marc Hallin, 2015. "Networks, Dynamic Factors, and the Volatility Analysis of High-Dimensional Financial Series," Working Papers ECARES ECARES 2015-34, ULB -- Universite Libre de Bruxelles.
    16. Miao, Ke & Phillips, Peter C.B. & Su, Liangjun, 2023. "High-dimensional VARs with common factors," Journal of Econometrics, Elsevier, vol. 233(1), pages 155-183.
    17. Narayan, Seema & Smyth, Russell, 2015. "The financial econometrics of price discovery and predictability," International Review of Financial Analysis, Elsevier, vol. 42(C), pages 380-393.
    18. Dai, Chaoxing & Lu, Kun & Xiu, Dacheng, 2019. "Knowing factors or factor loadings, or neither? Evaluating estimators of large covariance matrices with noisy and asynchronous data," Journal of Econometrics, Elsevier, vol. 208(1), pages 43-79.
    19. Ali Aghamohammadi, 2018. "Bayesian analysis of dynamic panel data by penalized quantile regression," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(1), pages 91-108, March.
    20. Xiaowei Zhang & Peter W. Glynn, 2018. "Affine Jump-Diffusions: Stochastic Stability and Limit Theorems," Papers 1811.00122, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sistpr:v:25:y:2022:i:2:d:10.1007_s11203-021-09257-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.