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A Consistent Bootstrap Test for Conditional Density Functions with Time-Dependent Data

Author

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  • Fuchun Li
  • Greg Tkacz

Abstract

This paper describes a new test for evaluating conditional density functions that remains valid when the data are time-dependent and that is therefore applicable to forecasting problems. We show that the test statistic is asymptotically distributed standard normal under the null hypothesis, and diverges to infinity when the null hypothesis is false. We use a bootstrap algorithm to approximate the distribution of the test statistic in finite samples, and show that the bootstrapped distribution converges to the asymptotic distribution in probability. A Monte Carlo simulation study reveals that the bootstrap test works well and is highly robust to the value of the smoothing parameter in the kernel density estimator. An application to inflation forecasting is also presented to demonstrate the use of the test.

Suggested Citation

  • Fuchun Li & Greg Tkacz, 2001. "A Consistent Bootstrap Test for Conditional Density Functions with Time-Dependent Data," Staff Working Papers 01-21, Bank of Canada.
  • Handle: RePEc:bca:bocawp:01-21
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    References listed on IDEAS

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    1. 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.
    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. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    4. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
    5. Pagan,Adrian & Ullah,Aman, 1999. "Nonparametric Econometrics," Cambridge Books, Cambridge University Press, number 9780521355643, Fall.
    6. Donald W. K. Andrews, 1997. "A Conditional Kolmogorov Test," Econometrica, Econometric Society, vol. 65(5), pages 1097-1128, September.
    7. Horowitz, Joel L., 1994. "Bootstrap-based critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395-411, April.
    8. 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.
    9. repec:cup:etheor:v:10:y:1994:i:2:p:316-56 is not listed on IDEAS
    10. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    11. Masry, Elias & Tjøstheim, Dag, 1995. "Nonparametric Estimation and Identification of Nonlinear ARCH Time Series Strong Convergence and Asymptotic Normality: Strong Convergence and Asymptotic Normality," Econometric Theory, Cambridge University Press, vol. 11(02), pages 258-289, February.
    12. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
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    Citations

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    Cited by:

    1. Giorgio Valente & Lucio Sarno, 2004. "Comparing the accuracy of density forecasts from competing models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(8), pages 541-557.
    2. Polanski, Arnold & Stoja, Evarist, 2012. "Efficient evaluation of multidimensional time-varying density forecasts, with applications to risk management," International Journal of Forecasting, Elsevier, vol. 28(2), pages 343-352.
    3. Corradi, Valentina & Swanson, Norman R., 2006. "Bootstrap conditional distribution tests in the presence of dynamic misspecification," Journal of Econometrics, Elsevier, vol. 133(2), pages 779-806, August.
    4. E. Fe-Rodríguez & C. Orme, 2006. "On the sensitivity of Kernel-based Conditional Moment Tests to Unconsidered Local Alternatives," The School of Economics Discussion Paper Series 0606, Economics, The University of Manchester.
    5. Hall, Stephen G. & Mitchell, James, 2007. "Combining density forecasts," International Journal of Forecasting, Elsevier, vol. 23(1), pages 1-13.
    6. Valentina Corradi & Norman R. Swanson, 2003. "A Test for Comparing Multiple Misspecified Conditional Distributions," Departmental Working Papers 200314, Rutgers University, Department of Economics.
    7. James Mitchell & Stephen G. Hall, 2005. "Evaluating, Comparing and Combining Density Forecasts Using the KLIC with an Application to the Bank of England and NIESR 'Fan' Charts of Inflation," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 67(s1), pages 995-1033, December.

    More about this item

    Keywords

    Econometric and statistical methods;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications

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