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