The Jacobian of the exponential function

Jan R. Magnus, Henk G.J. Pijls, Enrique Sentana*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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We derive closed-form expressions for the Jacobian of the matrix exponential function for both diagonalizable and defective matrices. The results are applied to two cases of interest in macroeconometrics: a continuous-time macro model and the parameterization of rotation matrices governing impulse response functions in structural vector autoregressions.

Original languageEnglish
Article number104122
Pages (from-to)1-15
Number of pages15
JournalJournal of Economic Dynamics and Control
Early online date8 May 2021
Publication statusPublished - Jun 2021

Bibliographical note

Funding Information:
We are grateful to the co-editor and one referee for positive and constructive feedback, and to Karim Abadir, Gabriele Fiorentini, Tom Koornwinder, Oliver Linton, Roderick McCrorie, Peter Phillips, and Peter Zadrozny for helpful comments and discussions. Sentana gratefully acknowledges financial support from the Spanish Ministry of Economy, Industry and Competitiveness through grant ECO 2017-89689. This research did not receive any other grant from funding agencies in the public, commercial, or not-for-profit sectors.

Publisher Copyright:
© 2021 Elsevier B.V.

Copyright 2021 Elsevier B.V., All rights reserved.


  • Continuous-time Markov chain
  • Impulse response analysis
  • Matrix differential calculus
  • Ornstein–Uhlenbeck process
  • Orthogonal matrix


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