IDEAS home Printed from https://ideas.repec.org/p/cdl/ucsdec/qt7g23r5kh.html
   My bibliography  Save this paper

Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis

Author

Listed:
  • Tanaka, Ken'ichiro
  • Toda, Alexis Akira

Abstract

The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback-Leibler information (relative entropy) of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints. We study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases. We prove that (i) the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with, and (ii) the approximate discrete distribution weakly converges to the given continuous distribution. Moreover, we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Tanaka, Ken'ichiro & Toda, Alexis Akira, 2015. "Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis," University of California at San Diego, Economics Working Paper Series qt7g23r5kh, Department of Economics, UC San Diego.
    • Handle: RePEc:cdl:ucsdec:qt7g23r5kh
    as

    Download full text from publisher

    File URL: https://www.escholarship.org/uc/item/7g23r5kh.pdf;origin=repeccitec
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Alexis Toda, 2015. "Bayesian general equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(2), pages 375-411, February.
    2. 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.
    3. Buchen, Peter W. & Kelly, Michael, 1996. "The Maximum Entropy Distribution of an Asset Inferred from Option Prices," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(1), pages 143-159, March.
    4. Foley Duncan K., 1994. "A Statistical Equilibrium Theory of Markets," Journal of Economic Theory, Elsevier, vol. 62(2), pages 321-345, April.
    5. Jerome Adda & Russell W. Cooper, 2003. "Dynamic Economics: Quantitative Methods and Applications," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012014, December.
    6. Tauchen, George, 1986. "Finite state markov-chain approximations to univariate and vector autoregressions," Economics Letters, Elsevier, vol. 20(2), pages 177-181.
    7. 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.
    8. Yuichi Kitamura & Michael Stutzer, 1997. "An Information-Theoretic Alternative to Generalized Method of Moments Estimation," Econometrica, Econometric Society, vol. 65(4), pages 861-874, July.
    9. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    10. David G. Luenberger & Yinyu Ye, 2008. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 0, number 978-0-387-74503-9, February.
    11. Stutzer, Michael, 1996. "A Simple Nonparametric Approach to Derivative Security Valuation," Journal of Finance, American Finance Association, vol. 51(5), pages 1633-1652, December.
    12. Wu, Ximing, 2003. "Calculation of maximum entropy densities with application to income distribution," Journal of Econometrics, Elsevier, vol. 115(2), pages 347-354, August.
    13. Donald L. Keefer & Samuel E. Bodily, 1983. "Three-Point Approximations for Continuous Random Variables," Management Science, INFORMS, vol. 29(5), pages 595-609, May.
    14. Alexis Toda, 2010. "Existence of a statistical equilibrium for an economy with endogenous offer sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(3), pages 379-415, December.
    15. Allen C. Miller, III & Thomas R. Rice, 1983. "Discrete Approximations of Probability Distributions," Management Science, INFORMS, vol. 29(3), pages 352-362, March.
    16. Paul A. Samuelson, 2011. "Lifetime Portfolio Selection by Dynamic Stochastic Programming," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 31, pages 465-472, World Scientific Publishing Co. Pte. Ltd..
    17. James E. Smith, 1993. "Moment Methods for Decision Analysis," Management Science, INFORMS, vol. 39(3), pages 340-358, March.
    18. Nikolay Gospodinov & Damba Lkhagvasuren, 2014. "A Moment‐Matching Method For Approximating Vector Autoregressive Processes By Finite‐State Markov Chains," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 29(5), pages 843-859, August.
    19. Kjetil Høyland & Stein W. Wallace, 2001. "Generating Scenario Trees for Multistage Decision Problems," Management Science, INFORMS, vol. 47(2), pages 295-307, February.
    20. 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.
    21. Tanaka, Ken’ichiro & Toda, Alexis Akira, 2013. "Discrete approximations of continuous distributions by maximum entropy," Economics Letters, Elsevier, vol. 118(3), pages 445-450.
    22. Stutzer, Michael, 1995. "A Bayesian approach to diagnosis of asset pricing models," Journal of Econometrics, Elsevier, vol. 68(2), pages 367-397, August.
    23. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. García, Benjamín & Giarda, Mario & Lizama, Carlos & Rojas, Ignacio, 2024. "Transmission mechanisms in HANK: An application to Chile," Latin American Journal of Central Banking (previously Monetaria), Elsevier, vol. 5(3).
    2. Lee, Ji Hyung & Sasaki, Yuya & Toda, Alexis Akira & Wang, Yulong, 2024. "Tuning parameter-free nonparametric density estimation from tabulated summary data," Journal of Econometrics, Elsevier, vol. 238(1).
    3. Roulleau-Pasdeloup, Jordan, 2023. "Analyzing Linear DSGE models: the Method of Undetermined Markov States," Journal of Economic Dynamics and Control, Elsevier, vol. 151(C).
    4. Leland E. Farmer & Alexis Akira Toda, 2017. "Discretizing nonlinear, non‐Gaussian Markov processes with exact conditional moments," Quantitative Economics, Econometric Society, vol. 8(2), pages 651-683, July.
    5. Alessandro Barbiero & Asmerilda Hitaj, 2023. "Discrete approximations of continuous probability distributions obtained by minimizing Cramér-von Mises-type distances," Statistical Papers, Springer, vol. 64(5), pages 1669-1697, October.
    6. Ma, Qingyin & Toda, Alexis Akira, 2022. "Asymptotic linearity of consumption functions and computational efficiency," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    7. Alexis Akira Toda, 2021. "Data-Based Automatic Discretization of Nonparametric Distributions," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 1217-1235, April.
    8. Fabrice Tourre, 2017. "A Macro-Finance Approach to Sovereign Debt Spreads and Returns," 2017 Meeting Papers 13, Society for Economic Dynamics.
    9. Gordon, Grey, 2021. "Efficient VAR discretization," Economics Letters, Elsevier, vol. 204(C).
    10. Ma, Qingyin & Toda, Alexis Akira, 2021. "A theory of the saving rate of the rich," Journal of Economic Theory, Elsevier, vol. 192(C).
    11. Robert Kirkby, 2025. "Discretizing earnings dynamics: implications of Gaussian-mixture shocks for life-cycle models," The Japanese Economic Review, Springer, vol. 76(2), pages 493-519, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lee, Ji Hyung & Sasaki, Yuya & Toda, Alexis Akira & Wang, Yulong, 2024. "Tuning parameter-free nonparametric density estimation from tabulated summary data," Journal of Econometrics, Elsevier, vol. 238(1).
    2. Tanaka, Ken’ichiro & Toda, Alexis Akira, 2013. "Discrete approximations of continuous distributions by maximum entropy," Economics Letters, Elsevier, vol. 118(3), pages 445-450.
    3. Jangho Yang, 2018. "Information Theoretic Approaches In Economics," Journal of Economic Surveys, Wiley Blackwell, vol. 32(3), pages 940-960, July.
    4. Adrien Auclert, 2019. "Monetary Policy and the Redistribution Channel," American Economic Review, American Economic Association, vol. 109(6), pages 2333-2367, June.
    5. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    6. Roulleau-Pasdeloup, Jordan, 2023. "Analyzing Linear DSGE models: the Method of Undetermined Markov States," Journal of Economic Dynamics and Control, Elsevier, vol. 151(C).
    7. Jess Benhabib & Alberto Bisin & Shenghao Zhu, 2011. "The Distribution of Wealth and Fiscal Policy in Economies With Finitely Lived Agents," Econometrica, Econometric Society, vol. 79(1), pages 123-157, January.
    8. Gordon, Grey, 2021. "Efficient VAR discretization," Economics Letters, Elsevier, vol. 204(C).
    9. Leland E. Farmer, 2021. "The discretization filter: A simple way to estimate nonlinear state space models," Quantitative Economics, Econometric Society, vol. 12(1), pages 41-76, January.
    10. Robert Kirkby, 2023. "Quantitative Macroeconomics: Lessons Learned from Fourteen Replications," Computational Economics, Springer;Society for Computational Economics, vol. 61(2), pages 875-896, February.
    11. Matteo Iacoviello, 2008. "Household Debt and Income Inequality, 1963–2003," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 40(5), pages 929-965, August.
    12. Andreas Fagereng & Luigi Guiso & Davide Malacrino & Luigi Pistaferri, 2020. "Heterogeneity and Persistence in Returns to Wealth," Econometrica, Econometric Society, vol. 88(1), pages 115-170, January.
    13. Ma Eunseong & Park Kwangyong, 2025. "Gini in the Taylor Rule: Should the Fed Care About Inequality?," The B.E. Journal of Macroeconomics, De Gruyter, vol. 25(1), pages 241-285.
    14. Takefumi Yamazaki, 2018. "Accuracy and speed of the solution methods for sovereign default models: The stable performance of the Tauchen method and cubic spline interpolation," Public Policy Review, Policy Research Institute, Ministry of Finance Japan, vol. 14(4), pages 641-662, July.
    15. Fernández-Villaverde, Jesús & Krueger, Dirk, 2011. "Consumption And Saving Over The Life Cycle: How Important Are Consumer Durables?," Macroeconomic Dynamics, Cambridge University Press, vol. 15(5), pages 725-770, November.
    16. Toda, Alexis Akira, 2019. "Wealth distribution with random discount factors," Journal of Monetary Economics, Elsevier, vol. 104(C), pages 101-113.
    17. Chipeniuk, Karsten O. & Katz, Nets Hawk & Walker, Todd B., 2022. "Households, auctioneers, and aggregation," European Economic Review, Elsevier, vol. 141(C).
    18. Eva Carceles-Poveda, 2009. "Asset Prices and Business Cycles under Market Incompleteness," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 12(3), pages 405-422, July.
    19. Conesa, Juan Carlos & Krueger, Dirk, 2006. "On the optimal progressivity of the income tax code," Journal of Monetary Economics, Elsevier, vol. 53(7), pages 1425-1450, October.
    20. Tu, Teng-Tsai, 1998. "An entropic approach to equity market integration and consumption-based capital asset pricing models," ISU General Staff Papers 1998010108000012895, Iowa State University, Department of Economics.

    More about this item

    Keywords

    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cdl:ucsdec:qt7g23r5kh. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Lisa Schiff (email available below). General contact details of provider: https://edirc.repec.org/data/deucsus.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.