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A Nonlinear Certainty Equivalent Approximation Method for Dynamic Stochastic Problems

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  • Yongyang Cai
  • Kenneth Judd
  • Jevgenijs Steinbuks

Abstract

This paper introduces a nonlinear certainty equivalent approximation method for dynamic stochastic problems. We first use a novel, stable and efficient method for computing the optimal policy functions for deterministic dynamic optimization problems, and then use them as certainty-equivalent approximations for the stochastic versions. Our examples demonstrate that it can be applied to solve high-dimensional problems with up to four hundred state variables with an acceptable accuracy. This method can also be applied to solve problems with inequality constraints that occasionally bind. These features make the nonlinear certainty equivalent approximation method suitable for solving complex economic problems, where other algorithms, such as log-linearization, fail or are far less tractable.

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  • Yongyang Cai & Kenneth Judd & Jevgenijs Steinbuks, 2015. "A Nonlinear Certainty Equivalent Approximation Method for Dynamic Stochastic Problems," NBER Working Papers 21590, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberwo:21590
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    1. 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.
    2. Jevgenijs Steinbuks & Thomas Hertel, 2016. "Confronting the Food–Energy–Environment Trilemma: Global Land Use in the Long Run," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 63(3), pages 545-570, March.
    3. Juillard, Michel & Villemot, Sébastien, 2011. "Multi-country real business cycle models: Accuracy tests and test bench," Journal of Economic Dynamics and Control, Elsevier, vol. 35(2), pages 178-185, February.
    4. Guerrieri, Luca & Iacoviello, Matteo, 2015. "OccBin: A toolkit for solving dynamic models with occasionally binding constraints easily," Journal of Monetary Economics, Elsevier, vol. 70(C), pages 22-38.
    5. Tauchen, George, 1986. "Finite state markov-chain approximations to univariate and vector autoregressions," Economics Letters, Elsevier, vol. 20(2), pages 177-181.
    6. 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.
    7. Ujjayant Chakravorty & Marie‐Hélène Hubert & Michel Moreaux & Linda Nøstbakken, 2017. "Long‐Run Impact of Biofuels on Food Prices," Scandinavian Journal of Economics, Wiley Blackwell, vol. 119(3), pages 733-767, July.
    8. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    9. Chakravorty, Ujjayant & Magné, Bertrand & Moreaux, Michel, 2008. "A dynamic model of food and clean energy," Journal of Economic Dynamics and Control, Elsevier, vol. 32(4), pages 1181-1203, April.
    10. Jordi Galí, 2008. "Introduction to Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework," Introductory Chapters, in: Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Princeton University Press.
    11. Del Negro, Marco & Schorfheide, Frank & Smets, Frank & Wouters, Rafael, 2007. "On the Fit of New Keynesian Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 123-143, April.
    12. Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
    13. Den Haan, Wouter J. & Judd, Kenneth L. & Juillard, Michel, 2011. "Computational suite of models with heterogeneous agents II: Multi-country real business cycle models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(2), pages 175-177, February.
    14. Benigno, Pierpaolo & Woodford, Michael, 2012. "Linear-quadratic approximation of optimal policy problems," Journal of Economic Theory, Elsevier, vol. 147(1), pages 1-42.
    15. Yongyang Cai & Kenneth L. Judd & Rong Xu, 2013. "Numerical Solution of Dynamic Portfolio Optimization with Transaction Costs," NBER Working Papers 18709, National Bureau of Economic Research, Inc.
    16. 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.
    17. Frank Smets & Rafael Wouters, 2007. "Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach," American Economic Review, American Economic Association, vol. 97(3), pages 586-606, June.
    18. Andreoni, James & Levinson, Arik, 2001. "The simple analytics of the environmental Kuznets curve," Journal of Public Economics, Elsevier, vol. 80(2), pages 269-286, May.
    19. Yongyang Cai & Kenneth Judd, 2013. "Shape-preserving dynamic programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 407-421, June.
    20. 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.
    21. 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.
    22. Lars Peter Hansen & Thomas J. Sargent, 2005. "Certainty equivalence and model uncertainty," Proceedings, Board of Governors of the Federal Reserve System (U.S.), pages 17-38.
    23. Yongyang Cai & Kenneth Judd, 2015. "Dynamic programming with Hermite approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 245-267, June.
    24. 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.
    25. Xepapadeas, Anastasios, 2005. "Economic growth and the environment," Handbook of Environmental Economics, in: K. G. Mäler & J. R. Vincent (ed.), Handbook of Environmental Economics, edition 1, volume 3, chapter 23, pages 1219-1271, Elsevier.
    26. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    27. Gaspar, Jess & L. Judd, Kenneth, 1997. "Solving Large-Scale Rational-Expectations Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(1), pages 45-75, January.
    28. Cai, Yongyang & Steinbuks, Jevgenijs & Elliott, Joshua & Hertel, Thomas W., 2014. "The effect of climate and technological uncertainty in crop yields on the optimal path of global land use," Policy Research Working Paper Series 7009, The World Bank.
    29. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
    30. Johannes Brumm & Simon Scheidegger, 2017. "Using Adaptive Sparse Grids to Solve High‐Dimensional Dynamic Models," Econometrica, Econometric Society, vol. 85, pages 1575-1612, September.
    31. Den Haan, Wouter J. & De Wind, Joris, 2012. "Nonlinear and stable perturbation-based approximations," Journal of Economic Dynamics and Control, Elsevier, vol. 36(10), pages 1477-1497.
    32. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    33. Dario Caldara & Jesus Fernandez-Villaverde & Juan Rubio-Ramirez & Wen Yao, 2012. "Computing DSGE Models with Recursive Preferences and Stochastic Volatility," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(2), pages 188-206, April.
    34. Taylor, John B., 1993. "Discretion versus policy rules in practice," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 39(1), pages 195-214, December.
    35. Cai, Yongyang & Judd, Kenneth L., 2012. "Dynamic programming with shape-preserving rational spline Hermite interpolation," Economics Letters, Elsevier, vol. 117(1), pages 161-164.
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    2. Haktanır, Elif & Kahraman, Cengiz, 2023. "Intuitionistic fuzzy risk adjusted discount rate and certainty equivalent methods for risky projects," International Journal of Production Economics, Elsevier, vol. 257(C).
    3. Yongyang Cai & Kenneth L. Judd, 2023. "A simple but powerful simulated certainty equivalent approximation method for dynamic stochastic problems," Quantitative Economics, Econometric Society, vol. 14(2), pages 651-687, May.
    4. Yongyang Cai, 2020. "The Role of Uncertainty in Controlling Climate Change," Papers 2003.01615, arXiv.org, revised Oct 2020.
    5. Duong Ngotran, 2016. "The E-Monetary Theory," 2016 Papers png175, Job Market Papers.
    6. Aldrich Eric Mark & Kung Howard, 2021. "Computational Methods for Production-Based Asset Pricing Models with Recursive Utility," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 25(1), pages 1-26, February.
    7. Jan Bruha, 2015. "Dynamics of Linear Forward-looking Structural Macroeconomic Models at the Zero Lower Bound: Do Solution Techniques Matter?," Working Papers 2015/13, Czech National Bank.
    8. Cai,Yongyang & Steinbuks,Jevgenijs & Judd,Kenneth L. & Jaegermeyr,Jonas & Hertel,Thomas W., 2020. "Modeling Uncertainty in Large Natural Resource Allocation Problems," Policy Research Working Paper Series 9159, The World Bank.
    9. Wonjun Chang & Michael C. Ferris & Youngdae Kim & Thomas F. Rutherford, 2020. "Solving Stochastic Dynamic Programming Problems: A Mixed Complementarity Approach," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 925-955, March.
    10. Golub, Alla & Cai, Yongyang & Hertel, Thomas & Steinbuks, Jevgenijs, 2015. "Energy Price Uncertainty and Global Land Use," 2015 AAEA & WAEA Joint Annual Meeting, July 26-28, San Francisco, California 205406, Agricultural and Applied Economics Association.
    11. Dennis, Richard, 2024. "Using a hyperbolic cross to solve non-linear macroeconomic models," Journal of Economic Dynamics and Control, Elsevier, vol. 163(C).

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • 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
    • E31 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Price Level; Inflation; Deflation
    • E52 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit - - - Monetary Policy

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