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Second-order approximation of dynamic models without the use of tensors

Several approaches to finding the second-order approximation to a dynamic model have been proposed recently. This paper differs from the existing literature in that it makes use of the Magnus and Neudecker (1999) definition of the Hessian matrix. The key result is a linear system of equations that characterizes the second-order coefficients. No use is made of multi-dimensional arrays or tensors, a practical implication of which is that it is much easier to transcribe the mathematical representation of the solution into usable computer code. Matlab code is available from; Fortran 90 code is available from

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Paper provided by Concordia University, Department of Economics in its series Working Papers with number 09004.

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Length: 27 pages
Date of creation: 17 Feb 2009
Date of revision: 28 Apr 2010
Handle: RePEc:crd:wpaper:09004
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  1. 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.
  2. OndŘej KamenÍk, 2005. "Solving SDGE Models: A New Algorithm for the Sylvester Equation," Computational Economics, Society for Computational Economics, vol. 25(1), pages 167-187, February.
  3. Tauchen, George & Hussey, Robert, 1991. "Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models," Econometrica, Econometric Society, vol. 59(2), pages 371-96, March.
  4. Lawrence J. Christiano, 1998. "Solving Dynamic Equilibrium Models by a Method of Undetermined Coefficients," NBER Technical Working Papers 0225, National Bureau of Economic Research, Inc.
  5. Jinill Kim & Sunghyun Henry Kim & Ernst Schaumburg & Christopher A. Sims, 2003. "Calculating and using second order accurate solutions of discrete time dynamic equilibrium models," Finance and Economics Discussion Series 2003-61, Board of Governors of the Federal Reserve System (U.S.).
  6. Giovanni Lombardo & Alan Sutherland, 2005. " Computing Second-Order-Accurate Solutions for Rational Expectation Models Using Linear Solution Methods," CDMA Conference Paper Series 0504, Centre for Dynamic Macroeconomic Analysis.
  7. King, Robert G & Watson, Mark W, 2002. "System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations," Computational Economics, Society for Computational Economics, vol. 20(1-2), pages 57-86, October.
  8. Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-11, July.
  9. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, June.
  10. 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.
  11. Coleman, Wilbur John, II, 1990. "Solving the Stochastic Growth Model by Policy-Function Iteration," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(1), pages 27-29, January.
  12. King, Robert G & Plosser, Charles I & Rebelo, Sergio T, 2002. "Production, Growth and Business Cycles: Technical Appendix," Computational Economics, Society for Computational Economics, vol. 20(1-2), pages 87-116, October.
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