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Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain

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  • Judd, Kenneth L.
  • Maliar, Lilia
  • Maliar, Serguei
  • Valero, Rafael
Abstract
We show how to enhance the performance of a Smolyak method for solving dynamic economic models. First, we propose a more efficient implementation of the Smolyak method for interpolation, namely, we show how to avoid costly evaluations of repeated basis functions in the conventional Smolyak formula. Second, we extend the Smolyak method to include anisotropic constructions that allow us to target higher quality of approximation in some dimensions than in others. Third, we show how to effectively adapt the Smolyak hypercube to a solution domain of a given economic model. Finally, we argue that in large-scale economic applications, a solution algorithm based on Smolyak interpolation has substantially lower expense when it uses derivative-free fixed-point iteration instead of standard time iteration. In the context of one- and multi-agent optimal growth models, we find that the proposed modifications to the conventional Smolyak method lead to substantial increases in accuracy and speed.

Suggested Citation

  • Judd, Kenneth L. & Maliar, Lilia & Maliar, Serguei & Valero, Rafael, 2014. "Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 92-123.
    • Handle: RePEc:eee:dyncon:v:44:y:2014:i:c:p:92-123
    DOI: 10.1016/j.jedc.2014.03.003
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    1. Aruoba, S. Boragan & Fernandez-Villaverde, Jesus & Rubio-Ramirez, Juan F., 2006. "Comparing solution methods for dynamic equilibrium economies," Journal of Economic Dynamics and Control, Elsevier, vol. 30(12), pages 2477-2508, December.
    2. Christiano, Lawrence J. & Fisher, Jonas D. M., 2000. "Algorithms for solving dynamic models with occasionally binding constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 24(8), pages 1179-1232, July.
    3. Maliar, Serguei & Maliar, Lilia & Judd, Kenneth, 2011. "Solving the multi-country real business cycle model using ergodic set methods," Journal of Economic Dynamics and Control, Elsevier, vol. 35(2), pages 207-228, February.
    4. Fernández-Villaverde, Jesús & Gordon, Grey & Guerrón-Quintana, Pablo & Rubio-Ramírez, Juan F., 2015. "Nonlinear adventures at the zero lower bound," Journal of Economic Dynamics and Control, Elsevier, vol. 57(C), pages 182-204.
    5. Smith, A A, Jr, 1993. "Estimating Nonlinear Time-Series Models Using Simulated Vector Autoregressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(S), pages 63-84, Suppl. De.
    6. Kollmann, Robert & Maliar, Serguei & Malin, Benjamin A. & Pichler, Paul, 2011. "Comparison of solutions to the multi-country Real Business Cycle model," Journal of Economic Dynamics and Control, Elsevier, vol. 35(2), pages 186-202, February.
    7. Lilia Maliar, 2015. "Assessing gains from parallel computation on a supercomputer," Economics Bulletin, AccessEcon, vol. 35(1), pages 159-167.
    8. Marcet, Albert & Sargent, Thomas J, 1989. "Convergence of Least-Squares Learning in Environments with Hidden State Variables and Private Information," Journal of Political Economy, University of Chicago Press, vol. 97(6), pages 1306-1322, December.
    9. Tauchen, George & Hussey, Robert, 1991. "Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models," Econometrica, Econometric Society, vol. 59(2), pages 371-396, March.
    10. Miranda, Mario J & Helmberger, Peter G, 1988. "The Effects of Commodity Price Stabilization Programs," American Economic Review, American Economic Association, vol. 78(1), pages 46-58, March.
    11. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    12. Malin, Benjamin A. & Krueger, Dirk & Kubler, Felix, 2011. "Solving the multi-country real business cycle model using a Smolyak-collocation method," Journal of Economic Dynamics and Control, Elsevier, vol. 35(2), pages 229-239, February.
    13. Dirk Krueger & Felix Kubler, 2006. "Pareto-Improving Social Security Reform when Financial Markets are Incomplete!?," American Economic Review, American Economic Association, vol. 96(3), pages 737-755, June.
    14. Maliar, Lilia & Maliar, Serguei, 2013. "Envelope condition method versus endogenous grid method for solving dynamic programming problems," Economics Letters, Elsevier, vol. 120(2), pages 262-266.
    15. Kenneth L. Judd & Lilia Maliar & Serguei Maliar, 2011. "Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models," Quantitative Economics, Econometric Society, vol. 2(2), pages 173-210, July.
    16. Viktor Winschel & Markus Kr‰tzig, 2010. "Solving, Estimating, and Selecting Nonlinear Dynamic Models Without the Curse of Dimensionality," Econometrica, Econometric Society, vol. 78(2), pages 803-821, March.
    17. Kenneth Judd & Lilia Maliar & Serguei Maliar, 2012. "Merging simulation and projection approaches to solve high-dimensional problems," Working Papers. Serie AD 2012-20, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    18. John Rust, 1997. "Using Randomization to Break the Curse of Dimensionality," Econometrica, Econometric Society, vol. 65(3), pages 487-516, May.
    19. Ángel Gavilán & Juan A. Rojas, 2009. "Solving Portfolio Problems with the Smolyak-Parameterized Expectations Algorithm," Working Papers 0838, Banco de España.
    20. Krueger, Dirk & Kubler, Felix, 2004. "Computing equilibrium in OLG models with stochastic production," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1411-1436, April.
    21. Brian D. Wright & Jeffrey C. Williams, 1984. "The Welfare Effects of the Introduction of Storage," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 99(1), pages 169-192.
    22. Gaspar, Jess & L. Judd, Kenneth, 1997. "Solving Large-Scale Rational-Expectations Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(1), pages 45-75, January.
    23. S. Rao Aiyagari, 1994. "Uninsured Idiosyncratic Risk and Aggregate Saving," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(3), pages 659-684.
    24. Carroll, Christopher D., 2006. "The method of endogenous gridpoints for solving dynamic stochastic optimization problems," Economics Letters, Elsevier, vol. 91(3), pages 312-320, June.
    25. Per Krusell & Anthony A. Smith & Jr., 1998. "Income and Wealth Heterogeneity in the Macroeconomy," Journal of Political Economy, University of Chicago Press, vol. 106(5), pages 867-896, October.
    26. Anthony A. Smith, Jr., 1991. "Solving Stochastic Dynamic Programming Problems Using Rules Of Thumb," Working Paper 816, Economics Department, Queen's University.
    27. Grey Gordon, 2020. "Computing Dynamic Heterogeneous-Agent Economies: Tracking the Distribution," Economic Quarterly, Federal Reserve Bank of Richmond, issue 2Q, pages 61-95.
    28. Viktor Winschel, 2005. "Solving, Estimating and Selecting Nonlinear Dynamic Economic Models without the Curse of Dimensionality," GE, Growth, Math methods 0507014, University Library of Munich, Germany.
    29. Judd, Kenneth L., 1992. "Projection methods for solving aggregate growth models," Journal of Economic Theory, Elsevier, vol. 58(2), pages 410-452, December.
    30. Den Haan, Wouter J., 1990. "The optimal inflation path in a Sidrauski-type model with uncertainty," Journal of Monetary Economics, Elsevier, vol. 25(3), pages 389-409, June.
    31. Johannes Brumm & Simon Scheidegger, 2017. "Using Adaptive Sparse Grids to Solve High‐Dimensional Dynamic Models," Econometrica, Econometric Society, vol. 85, pages 1575-1612, September.
    32. Kenneth L. Judd & Lilia Maliar & Serguei Maliar, 2010. "A Cluster-Grid Projection Method: Solving Problems with High Dimensionality," NBER Working Papers 15965, National Bureau of Economic Research, Inc.
    33. Winschel, Viktor & Krätzig, Markus, 2008. "Solving, estimating and selecting nonlinear dynamic models without the curse of dimensionality," SFB 649 Discussion Papers 2008-018, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    34. Huggett, Mark, 1993. "The risk-free rate in heterogeneous-agent incomplete-insurance economies," Journal of Economic Dynamics and Control, Elsevier, vol. 17(5-6), pages 953-969.
    35. Marimon, Ramon & Scott, Andrew (ed.), 1999. "Computational Methods for the Study of Dynamic Economies," OUP Catalogue, Oxford University Press, number 9780198294979.
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    More about this item

    Keywords

    Smolyak method; Sparse grid; Adaptive domain; Projection; Anisotropic grid; High-dimensional problem;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C68 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models

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