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Quasi-likelihood analysis for Student-Lévy regression

Author

Listed:
  • Hiroki Masuda

    (University of Tokyo)

  • Lorenzo Mercuri

    (University of Milan)

  • Yuma Uehara

    (Kansai University)

Abstract

We consider the quasi-likelihood analysis for a linear regression model driven by a Student-t Lévy process with constant scale and arbitrary degrees of freedom. The model is observed at high frequency over an extending period, under which we can quantify how the sampling frequency affects estimation accuracy. In that setting, joint estimation of trend, scale, and degrees of freedom is a non-trivial problem. The bottleneck is that the Student-t distribution is not closed under convolution, making it difficult to estimate all the parameters fully based on the high-frequency time scale. To efficiently deal with the intricate nature from both theoretical and computational points of view, we propose a two-step quasi-likelihood analysis: first, we make use of the Cauchy quasi-likelihood for estimating the regression-coefficient vector and the scale parameter; then, we construct the sequence of the unit-period cumulative residuals to estimate the remaining degrees of freedom. In particular, using full data in the first step causes a problem stemming from the small-time Cauchy approximation, showing the need for data thinning.

Suggested Citation

  • Hiroki Masuda & Lorenzo Mercuri & Yuma Uehara, 2024. "Quasi-likelihood analysis for Student-Lévy regression," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 761-794, October.
    • Handle: RePEc:spr:sistpr:v:27:y:2024:i:3:d:10.1007_s11203-024-09317-2
    DOI: 10.1007/s11203-024-09317-2
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    References listed on IDEAS

    as
    1. Hiroki Masuda & Lorenzo Mercuri & Yuma Uehara, 2024. "Student t-L\'evy regression model in YUIMA," Papers 2403.12078, arXiv.org.
    2. Nakahiro Yoshida, 2011. "Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(3), pages 431-479, June.
    3. 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.
    4. Masuda, Hiroki, 2019. "Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 1013-1059.
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