IDEAS home Printed from https://ideas.repec.org/p/cmf/wpaper/wp2020_2005.html
   My bibliography  Save this paper

The Jacobian of the Exponential Function

Author

Listed:

Abstract

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 parametrization of rotation matrices governing impulse response functions in structural vector autoregressions.

Suggested Citation

  • Jan R. Magnus & Henk G. J. Pijls & Enrique Sentana, 2020. "The Jacobian of the Exponential Function," Working Papers wp2020_2005, CEMFI.
    • Handle: RePEc:cmf:wpaper:wp2020_2005
    as

    Download full text from publisher

    File URL: https://www.cemfi.es/ftp/wp/2005.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Giovanni Angelini & Emanuele Bacchiocchi & Giovanni Caggiano & Luca Fanelli, 2019. "Uncertainty across volatility regimes," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(3), pages 437-455, April.
    2. Gouriéroux, Christian & Monfort, Alain & Renne, Jean-Paul, 2017. "Statistical inference for independent component analysis: Application to structural VAR models," Journal of Econometrics, Elsevier, vol. 196(1), pages 111-126.
    3. Roy van der Weide, 2002. "GO-GARCH: a multivariate generalized orthogonal GARCH model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 549-564.
    4. Lanne, Markku & Meitz, Mika & Saikkonen, Pentti, 2017. "Identification and estimation of non-Gaussian structural vector autoregressions," Journal of Econometrics, Elsevier, vol. 196(2), pages 288-304.
    5. Weide, R. van der, 2002. "Generalized Orthogonal GARCH. A Multivariate GARCH model," CeNDEF Working Papers 02-02, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    6. Juan F. Rubio-Ramírez & Daniel F. Waggoner & Tao Zha, 2010. "Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(2), pages 665-696.
    7. Bergstrom, A.R., 1984. "Continuous time stochastic models and issues of aggregation over time," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 20, pages 1145-1212, Elsevier.
    8. Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2018. "Measuring Uncertainty and Its Impact on the Economy," The Review of Economics and Statistics, MIT Press, vol. 100(5), pages 799-815, December.
    9. Comte, F. & Renault, E., 1996. "Noncausality in Continuous Time Models," Econometric Theory, Cambridge University Press, vol. 12(2), pages 215-256, June.
    10. Kyle Jurado & Sydney C. Ludvigson & Serena Ng, 2015. "Measuring Uncertainty," American Economic Review, American Economic Association, vol. 105(3), pages 1177-1216, March.
    11. Hansen, Lars Peter & Sargent, Thomas J, 1983. "The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities," Econometrica, Econometric Society, vol. 51(2), pages 377-387, March.
    12. Kawakatsu, Hiroyuki, 2006. "Matrix exponential GARCH," Journal of Econometrics, Elsevier, vol. 134(1), pages 95-128, September.
    13. Chen, Baoline & Zadrozny, Peter A., 2001. "Analytic derivatives of the matrix exponential for estimation of linear continuous-time models1," Journal of Economic Dynamics and Control, Elsevier, vol. 25(12), pages 1867-1879, December.
    14. Linton, Oliver & McCrorie, J. Roderick, 1995. "Differentiation of an Exponential Matrix Function," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1182-1185, October.
    15. Markus Brunnermeier & Darius Palia & Karthik A. Sastry & Christopher A. Sims, 2021. "Feedbacks: Financial Markets and Economic Activity," American Economic Review, American Economic Association, vol. 111(6), pages 1845-1879, June.
    16. Phillips, P. C. B., 1973. "The problem of identification in finite parameter continuous time models," Journal of Econometrics, Elsevier, vol. 1(4), pages 351-362, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Lee, Adam & Mesters, Geert, 2024. "Locally robust inference for non-Gaussian linear simultaneous equations models," Journal of Econometrics, Elsevier, vol. 240(1).
    2. Dante Amengual & Gabriele Fiorentini & Enrique Sentana, 2022. "Moment tests of independent components," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 13(1), pages 429-474, May.
    3. Fiorentini, Gabriele & Sentana, Enrique, 2023. "Discrete mixtures of normals pseudo maximum likelihood estimators of structural vector autoregressions," Journal of Econometrics, Elsevier, vol. 235(2), pages 643-665.

    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. Hafner, Christian M. & Herwartz, Helmut & Maxand, Simone, 2022. "Identification of structural multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 227(1), pages 212-227.
    2. Cordoni, Francesco & Dorémus, Nicolas & Moneta, Alessio, 2024. "Identification of vector autoregressive models with nonlinear contemporaneous structure," Journal of Economic Dynamics and Control, Elsevier, vol. 162(C).
    3. Brianti, Marco, 2021. "Financial Shocks, Uncertainty Shocks, and Monetary Policy Trade-Offs," Working Papers 2021-5, University of Alberta, Department of Economics.
    4. Gabriele Fiorentini & Enrique Sentana, 2021. "Specification tests for non‐Gaussian maximum likelihood estimators," Quantitative Economics, Econometric Society, vol. 12(3), pages 683-742, July.
    5. Dominik Bertsche & Robin Braun, 2022. "Identification of Structural Vector Autoregressions by Stochastic Volatility," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 328-341, January.
    6. Chambers, MJ & McCrorie, JR & Thornton, MA, 2017. "Continuous Time Modelling Based on an Exact Discrete Time Representation," Economics Discussion Papers 20497, University of Essex, Department of Economics.
    7. Carriero, Andrea & Clark, Todd E. & Marcellino, Massimiliano, 2021. "Using time-varying volatility for identification in Vector Autoregressions: An application to endogenous uncertainty," Journal of Econometrics, Elsevier, vol. 225(1), pages 47-73.
    8. Han, Chulwoo & Park, Frank C., 2022. "A geometric framework for covariance dynamics," Journal of Banking & Finance, Elsevier, vol. 134(C).
    9. Asai, Manabu & McAleer, Michael, 2015. "Forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance," Journal of Econometrics, Elsevier, vol. 189(2), pages 251-262.
    10. Herwartz, Helmut & Lange, Alexander & Maxand, Simone, 2019. "Statistical identification in SVARs - Monte Carlo experiments and a comparative assessment of the role of economic uncertainties for the US business cycle," University of Göttingen Working Papers in Economics 375, University of Goettingen, Department of Economics.
    11. Manabu Asai & Michael McAleer, 2009. "Dynamic Conditional Correlations for Asymmetric Processes," CARF F-Series CARF-F-168, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    12. Josué Diwambuena & Jean-Paul K. Tsasa, 2021. "The Real Effects of Uncertainty Shocks: New Evidence from Linear and Nonlinear SVAR Models," BEMPS - Bozen Economics & Management Paper Series BEMPS87, Faculty of Economics and Management at the Free University of Bozen.
    13. Yu, Jun, 2014. "Econometric Analysis Of Continuous Time Models: A Survey Of Peter Phillips’S Work And Some New Results," Econometric Theory, Cambridge University Press, vol. 30(4), pages 737-774, August.
    14. Rivolta, Giulia & Trecroci, Carmine, 2020. "Measuring the effects of U.S. uncertainty and monetary conditions on EMEs' macroeconomic dynamics," MPRA Paper 99403, University Library of Munich, Germany.
    15. Caggiano, Giovanni & Castelnuovo, Efrem & Delrio, Silvia & Kima, Richard, 2021. "Financial uncertainty and real activity: The good, the bad, and the ugly," European Economic Review, Elsevier, vol. 136(C).
    16. Giovanni Angelini & Luca Fanelli, 2019. "Exogenous uncertainty and the identification of structural vector autoregressions with external instruments," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(6), pages 951-971, September.
    17. Helmut Herwartz & Alexander Lange, 2024. "How certain are we about the role of uncertainty in the economy?," Economic Inquiry, Western Economic Association International, vol. 62(1), pages 126-149, January.
    18. Michael Ryan, 2020. "A Narrative Approach to Creating Instruments with Unstructured and Voluminous Text: An Application to Policy Uncertainty," Working Papers in Economics 20/10, University of Waikato.
    19. Wang, Xiaohu & Phillips, Peter C.B. & Yu, Jun, 2011. "Bias in estimating multivariate and univariate diffusions," Journal of Econometrics, Elsevier, vol. 161(2), pages 228-245, April.
    20. Herwartz, Helmut & Wang, Shu, 2023. "Point estimation in sign-restricted SVARs based on independence criteria with an application to rational bubbles," Journal of Economic Dynamics and Control, Elsevier, vol. 151(C).

    More about this item

    Keywords

    Matrix differential calculus; orthogonal matrix; continuous-time Markov chain; Ornstein-Uhlenbeck process.;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:cmf:wpaper:wp2020_2005. 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: Araceli Requerey (email available below). General contact details of provider: https://edirc.repec.org/data/cemfies.html .

    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.