IDEAS home Printed from
   My bibliography  Save this paper

Density Estimation and Combination under Model Ambiguity


  • Stefania D'Amico


This paper proposes a method for estimating the probability density of a variable of interest in the presence of model ambiguity. In the first step, each candidate parametric model is estimated minimizing the Kullback-Leibler "distance" (KLD) from a reference nonparametric density estimate. Given that the KLD represents a measure of uncertainty about the true structure, in the second step, its information content is used to rank and combine the estimated models. The paper shows that the resulting parameters estimator is root-n consistent and asymptotically normally distributed. The KLD between the nonparametric and the parametric density estimates is also shown to be asymptotically normally distributed. This result leads to determining the weights in the model combination, using the distribution function of a Normal centered on the average performance of all plausible models. As such, this combination technique does not require the true structure to belong to the set of competing models and is computationally simple. I apply the proposed method to estimate the density function of daily stock returns under different phases of the business cycle. The results indicate that the double Gamma distribution is more adequate than the Gaussian distribution in modeling stock returns, and that the combination outperforms each individual candidate model both in- and out-of-sample

Suggested Citation

  • Stefania D'Amico, 2004. "Density Estimation and Combination under Model Ambiguity," Computing in Economics and Finance 2004 273, Society for Computational Economics.
  • Handle: RePEc:sce:scecf4:273

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Andrew Ang & Geert Bekaert, 2003. "How do Regimes Affect Asset Allocation?," NBER Working Papers 10080, National Bureau of Economic Research, Inc.
    2. Zheng, John Xu, 2000. "A Consistent Test Of Conditional Parametric Distributions," Econometric Theory, Cambridge University Press, vol. 16(05), pages 667-691, October.
    3. Itzhak Gilboa & David Schmeidler, 2003. "Inductive Inference: An Axiomatic Approach," Econometrica, Econometric Society, vol. 71(1), pages 1-26, January.
    4. Raffaella Giacomini & Halbert White, 2006. "Tests of Conditional Predictive Ability," Econometrica, Econometric Society, vol. 74(6), pages 1545-1578, November.
    5. Andrew Ang & Geert Bekaert, 2002. "International Asset Allocation With Regime Shifts," Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1137-1187.
    6. Sin, Chor-Yiu & White, Halbert, 1996. "Information criteria for selecting possibly misspecified parametric models," Journal of Econometrics, Elsevier, vol. 71(1-2), pages 207-225.
    7. Francis X. Diebold & Todd A. Gunther & Anthony S. Tay, "undated". "Evaluating Density Forecasts," CARESS Working Papres 97-18, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    8. Raffaella Giacomini, 2002. "Comparing Density Forecasts via Weighted Likelihood Ratio Tests: Asymptotic and Bootstrap Methods," Boston College Working Papers in Economics 583, Boston College Department of Economics.
    9. Doron Avramov, "undated". "Stock-Return Predictability and Model Uncertainty," Rodney L. White Center for Financial Research Working Papers 12-00, Wharton School Rodney L. White Center for Financial Research.
    10. L. Wade, 1988. "Review," Public Choice, Springer, vol. 58(1), pages 99-100, July.
    11. Raman Uppal & Tan Wang, 2003. "Model Misspecification and Underdiversification," Journal of Finance, American Finance Association, vol. 58(6), pages 2465-2486, December.
    12. Dhrymes, Phoebus J., 1998. "Identification and Kullback information in the GLSEM," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 163-184.
    13. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-560, May.
    14. David Schmeidler, 2000. "Cognitive Foundations of Inductive Inference and Probability: An Axiomatic Approach," Working Papers 00-07, Ohio State University, Department of Economics.
    15. repec:cup:etheor:v:10:y:1994:i:2:p:316-56 is not listed on IDEAS
    16. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
    17. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    18. Maasoumi, Esfandiar & Racine, Jeff, 2002. "Entropy and predictability of stock market returns," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 291-312, March.
    19. P. M. Robinson, 1991. "Consistent Nonparametric Entropy-Based Testing," Review of Economic Studies, Oxford University Press, vol. 58(3), pages 437-453.
    20. Fan, Yanqin, 1994. "Testing the Goodness of Fit of a Parametric Density Function by Kernel Method," Econometric Theory, Cambridge University Press, vol. 10(02), pages 316-356, June.
    21. Francis X. Diebold & Jose A. Lopez, 1995. "Forecast evaluation and combination," Research Paper 9525, Federal Reserve Bank of New York.
    22. David F. Hendry & Michael P. Clements, 2004. "Pooling of forecasts," Econometrics Journal, Royal Economic Society, vol. 7(1), pages 1-31, June.
    23. Yongmiao Hong & Halbert White, 2005. "Asymptotic Distribution Theory for Nonparametric Entropy Measures of Serial Dependence," Econometrica, Econometric Society, vol. 73(3), pages 837-901, May.
    24. K. J. Martijn Cremers, 2002. "Stock Return Predictability: A Bayesian Model Selection Perspective," Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1223-1249.
    25. John Xu Zheng, 1996. "A consistent test of functional form via nonparametric estimation techniques," Journal of Econometrics, Elsevier, vol. 75(2), pages 263-289, December.
    26. J. L. Knight & S. E. Satchell & K. C. Tran, 1995. "Statistical modelling of asymmetric risk in asset returns," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(3), pages 155-172.
    27. Lars Peter Hansen & Thomas J. Sargent, 2001. "Acknowledging Misspecification in Macroeconomic Theory," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(3), pages 519-535, July.
    28. Avramov, Doron, 2002. "Stock return predictability and model uncertainty," Journal of Financial Economics, Elsevier, vol. 64(3), pages 423-458, June.
    29. Ebrahimi, Nader & Maasoumi, Esfandiar & Soofi, Ehsan S., 1999. "Ordering univariate distributions by entropy and variance," Journal of Econometrics, Elsevier, vol. 90(2), pages 317-336, June.
    30. Sawa, Takamitsu, 1978. "Information Criteria for Discriminating among Alternative Regression Models," Econometrica, Econometric Society, vol. 46(6), pages 1273-1291, November.
    Full references (including those not matched with items on IDEAS)

    More about this item


    density forecasting; kullback-Leibler information; model combination;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:sce:scecf4:273. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.