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Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem


  • Baoline Chen


  • Peter A. Zadrozny

    () (Bureau of Labor Statistics, 2 Massachusetts Ave., NE, Washington, DC 20212, U.S.A.)


The paper obtains two principal results. First, using a new definition of higher-order (>2) matrix derivatives, the paper derives a recursion for computing any Gaussian multivariate moment. Second, the paper uses this result in a perturbation method to derive equations for computing the 4th-order Taylor-series approximation of the objective function of the linear-quadratic exponential Gaussian (LQEG) optimal control problem. Previously, Karp (1985) formulated the 4th multivariate Gaussian moment in terms of MacRae's definition of a matrix derivative. His approach extends with difficulty to any higher (>4) multivariate Gaussian moment. The present recursion straightforwardly computes any multivariate Gaussian moment. Karp used his formulation of the Gaussian 4th moment to compute a 2nd-order approximation of the finite-horizon LQEG objective function. Using the simpler formulation, the present paper applies the perturbation method to derive equations for computing a 4th-order approximation of the infinite-horizon LQEG objective function. By illustrating a convenient definition of matrix derivatives in the numerical solution of the LQEG problem with the perturbation method, the paper contributes to the computational economist's toolbox for solving stochastic nonlinear dynamic optimization problems.

Suggested Citation

  • Baoline Chen & Peter A. Zadrozny, 2003. "Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem," Computational Economics, Springer;Society for Computational Economics, vol. 21(1_2), pages 45-64, February.
  • Handle: RePEc:kap:compec:v:21:y:2003:i:1_2:p:45-64

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    References listed on IDEAS

    1. Collard, Fabrice & Juillard, Michel, 2001. "Accuracy of stochastic perturbation methods: The case of asset pricing models," Journal of Economic Dynamics and Control, Elsevier, vol. 25(6-7), pages 979-999, June.
    2. Karp, Larry S., 1985. "Higher moments in the linear-quadratic-gaussian problem," Journal of Economic Dynamics and Control, Elsevier, vol. 9(1), pages 41-54, September.
    3. Peter A. Zadrozny & Baoline Chen, 1999. "Perturbation Solution of Nonlinear Rational Expectations Models," Computing in Economics and Finance 1999 334, Society for Computational Economics.
    4. Baoline Chen & A. Zadrozny, 2000. "Estimated U.S. Manufacturing Capital And Productivity Based On An Estimated Dynamic Economic Model," Computing in Economics and Finance 2000 133, Society for Computational Economics.
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    Cited by:

    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. Baoline Chen & Peter A. Zadrozny, 2005. "Multi-Step Perturbation Solution of Nonlinear Rational Expectations Models," Computing in Economics and Finance 2005 254, Society for Computational Economics.
    3. Lan, Hong & Meyer-Gohde, Alexander, 2013. "Solving DSGE models with a nonlinear moving average," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2643-2667.
    4. Chen, Baoline & Zadrozny, Peter A., 2009. "Multi-step perturbation solution of nonlinear differentiable equations applied to an econometric analysis of productivity," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2061-2074, April.
    5. Andrew Binning, 2013. "Third-order approximation of dynamic models without the use of tensors," Working Paper 2013/13, Norges Bank.
    6. Anderson, Evan W. & Hansen, Lars Peter & Sargent, Thomas J., 2012. "Small noise methods for risk-sensitive/robust economies," Journal of Economic Dynamics and Control, Elsevier, vol. 36(4), pages 468-500.

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