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Efficient Estimation of Copula-based Semiparametric Markov Models

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    This paper considers efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions; where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functionals are root-$n$ consistent, asymptotically normal and efficient; and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. We present Monte Carlo studies to compare the finite sample performance of the sieve MLE, the two-step estimator of Chen and Fan (2006), the correctly specified parametric MLE and the incorrectly specified parametric MLE. The simulation results indicate that our sieve MLEs perform very well; having much smaller biases and smaller variances than the two-step estimator for Markov models generated via Clayton, Gumbel and other tail dependent copulas.

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    File URL: http://cowles.econ.yale.edu/P/cd/d16b/d1691.pdf
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    Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1691.

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    Length: 55 pages
    Date of creation: Feb 2009
    Date of revision: Mar 2009
    Publication status: Published in Annals of Statistics (2009), 37(6B): 4214-4253
    Handle: RePEc:cwl:cwldpp:1691
    Contact details of provider: Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
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    Web page: http://cowles.econ.yale.edu/

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    Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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    1. Xiaohong Chen & Roger Koenker & Zhijie Xiao, 2009. "Copula-based nonlinear quantile autoregression," Econometrics Journal, Royal Economic Society, vol. 12(s1), pages S50-S67, 01.
    2. Andrew Patton, 2004. "Modelling Asymmetric Exchange Rate Dependence," Working Papers wp04-04, Warwick Business School, Finance Group.
    3. Chen, Xiaohong & Fan, Yanqin & Tsyrennikov, Viktor, 2006. "Efficient Estimation of Semiparametric Multivariate Copula Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1228-1240, September.
    4. Chunrong Ai & Xiaohong Chen, 2003. "Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions," Econometrica, Econometric Society, vol. 71(6), pages 1795-1843, November.
    5. Chen, Jian & Peng, Liang & Zhao, Yichuan, 2009. "Empirical likelihood based confidence intervals for copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 137-151, January.
    6. Jianqing Fan & Jiancheng Jiang, 2007. "Nonparametric inference with generalized likelihood ratio tests," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 16(3), pages 409-444, December.
    7. Xiaohong Chen & Xiaotong Shen, 1998. "Sieve Extremum Estimates for Weakly Dependent Data," Econometrica, Econometric Society, vol. 66(2), pages 289-314, March.
    8. Nze, Patrick Ango & Doukhan, Paul, 2004. "Weak Dependence: Models And Applications To Econometrics," Econometric Theory, Cambridge University Press, vol. 20(06), pages 995-1045, December.
    9. repec:ner:tilbur:urn:nbn:nl:ui:12-86725 is not listed on IDEAS
    10. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    11. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation of copula-based semiparametric time series models," Journal of Econometrics, Elsevier, vol. 130(2), pages 307-335, February.
    12. Beare, Brendan, 2008. "Copulas and Temporal Dependence," University of California at San Diego, Economics Working Paper Series qt2880q2jq, Department of Economics, UC San Diego.
    13. Genest, C. & Werker, B.J.M., 2001. "Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models," Other publications TiSEM b733c3f4-38d2-49aa-a2c7-4, Tilburg University, School of Economics and Management.
    14. Koenker, Roger & Xiao, Zhijie, 2006. "Quantile Autoregression," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 980-990, September.
    15. Xiaotong Shen & Hsin-Cheng Huang & Jimmy Ye, 2004. "Inference After Model Selection," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 751-762, January.
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