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Solving second and third-order approximations to DSGE models: A recursive Sylvester equation solution

Author

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  • Andrew Binning

    (Norges Bank (Central Bank of Norway))

Abstract

In this paper I derive the matrix chain rules for solving a second and a third-order approximation to a DSGE model that allow the use of a recursive Sylvester equation solution method. In particular I use the solution algorithms of Kamenik (2005) and Martin & Van Loan (2006) to solve the generalised Sylvester equations. Because I use matrix algebra instead of tensor notation to find the system of equations, I am able to provide standalone Matlab routines that make it feasible to solve a medium scale DSGE model in a competitive time. I also provide Fortran code and Matlab/Fortran mex files for my method.

Suggested Citation

  • Andrew Binning, 2013. "Solving second and third-order approximations to DSGE models: A recursive Sylvester equation solution," Working Paper 2013/18, Norges Bank.
    • Handle: RePEc:bno:worpap:2013_18
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    File URL: https://www.norges-bank.no/en/news-events/news-publications/Papers/Working-Papers/2013/WP-201318/
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    References listed on IDEAS

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    1. Klein, Paul, 2000. "Using the generalized Schur form to solve a multivariate linear rational expectations model," Journal of Economic Dynamics and Control, Elsevier, vol. 24(10), pages 1405-1423, September.
    2. Gali­, Jordi & Monacelli, Tommaso, 2008. "Optimal monetary and fiscal policy in a currency union," Journal of International Economics, Elsevier, vol. 76(1), pages 116-132, September.
    3. OndŘej KamenÍk, 2005. "Solving SDGE Models: A New Algorithm for the Sylvester Equation," Computational Economics, Springer;Society for Computational Economics, vol. 25(1), pages 167-187, February.
    4. Schmitt-Grohe, Stephanie & Uribe, Martin, 2004. "Solving dynamic general equilibrium models using a second-order approximation to the policy function," Journal of Economic Dynamics and Control, Elsevier, vol. 28(4), pages 755-775, January.
    5. RUGE-MURCIA, Francisco J., 2010. "Estimating Nonlinear DSGE Models by the Simulated Method of Moments," Cahiers de recherche 19-2010, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    6. Andrew Binning, 2013. "Third-order approximation of dynamic models without the use of tensors," Working Paper 2013/13, Norges Bank.
    7. Martin Andreasen, 2012. "On the Effects of Rare Disasters and Uncertainty Shocks for Risk Premia in Non-Linear DSGE Models," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(3), pages 295-316, July.
    8. Gomme, Paul & Klein, Paul, 2011. "Second-order approximation of dynamic models without the use of tensors," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 604-615, April.
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    Citations

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    Cited by:

    1. Parra-Alvarez, Juan Carlos & Polattimur, Hamza & Posch, Olaf, 2021. "Risk matters: Breaking certainty equivalence in linear approximations," Journal of Economic Dynamics and Control, Elsevier, vol. 133(C).
    2. Andreasen, Martin Møller & Caggiano, Giovanni & Castelnuovo, Efrem & Pellegrino, Giovanni, 2021. "Why does risk matter more in recessions than in expansions?," Bank of Finland Research Discussion Papers 13/2021, Bank of Finland.
    3. Junior Maih, 2014. "Efficient Perturbation Methods for Solving Regime-Switching DSGE Models," Working Papers No 10/2014, Centre for Applied Macro- and Petroleum economics (CAMP), BI Norwegian Business School.
    4. Levintal, Oren, 2017. "Fifth-order perturbation solution to DSGE models," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 1-16.
    5. Martin M. Andreasen & Anders Kronborg, 2017. "The Extended Perturbation Method: New Insights on the New Keynesian Model," CREATES Research Papers 2017-14, Department of Economics and Business Economics, Aarhus University.
    6. repec:zbw:bofrdp:2021_013 is not listed on IDEAS
    7. Martin M. Andreasen, 2019. "Explaining Bond Return Predictability in an Estimated New Keynesian Model," CREATES Research Papers 2019-11, Department of Economics and Business Economics, Aarhus University.
    8. Martin M. Andreasen & Giovanni Caggiano & Efrem Castelnuovo & Giovanni Pellegrino, 2021. "Why Does Risk Matter More in Recessions than in Expansions?," "Marco Fanno" Working Papers 0275, Dipartimento di Scienze Economiche "Marco Fanno".
    9. Martin M. Andreasen & Kasper Jørgensen, 2016. "Explaining Asset Prices with Low Risk Aversion and Low Intertemporal Substitution," CREATES Research Papers 2016-16, Department of Economics and Business Economics, Aarhus University.

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    Keywords

    Solving dynamic models; Second-order approximation; Third-order appeoximation; Second-order matrix chain rule; Third-order matrix chain rule; Generalised Sylvester equations;
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