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Estimating the Error Distribution in the Multivariate Heteroscedastic Time Series Models


  • Gunky Kim


  • Mervyn J. Silvapulle


  • Paramsothy Silvapulle



A semiparametric method is studied for estimating the dependence parameter and the joint distribution of the error term in a class of multivariate time series models when the marginal distributions of the errors are unknown. This method is a natural extension of Genest et al. (1995a) for independent and identically distributed observations. The proposed method first obtains √n-consistent estimates of the parameters of each univariate marginal time-series, and computes the corresponding residuals. These are then used to estimate the joint distribution of the multivariate error terms, which is specified using a copula. Our developments and proofs make use of, and build upon, recent elegant results of Koul and Ling (2006) and Koul (2002) for these models. The rigorous proofs provided here also lay the foundation and collect together the technical arguments that would be useful for other potential extensions of this semiparametric approach. It is shown that the proposed estimator of the dependence parameter of the multivariate error term is asymptotically normal, and a consistent estimator of its large sample variance is also given so that confidence intervals may be constructed. A large scale simulation study was carried out to compare the estimators particularly when the error distributions are unknown, which is almost always the case in practice. In this simulation study, our proposed semiparametric method performed better than the well-known parametric methods. An example on exchange rates is used to illustrate the method.

Suggested Citation

  • Gunky Kim & Mervyn J. Silvapulle & Paramsothy Silvapulle, 2007. "Estimating the Error Distribution in the Multivariate Heteroscedastic Time Series Models," Monash Econometrics and Business Statistics Working Papers 8/07, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2007-8

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    References listed on IDEAS

    1. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 125-154.
    2. Weijing Wang, 2003. "Estimating the association parameter for copula models under dependent censoring," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 257-273.
    3. David Oakes, 2003. "Copula model generated by Dabrowska's association measure," Biometrika, Biometrika Trust, vol. 90(2), pages 478-481, June.
    4. Kim, Gunky & Silvapulle, Mervyn J. & Silvapulle, Paramsothy, 2007. "Comparison of semiparametric and parametric methods for estimating copulas," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 2836-2850, March.
    5. Andrew J. Patton, 2006. "Modelling Asymmetric Exchange Rate Dependence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(2), pages 527-556, May.
    6. Markus Junker & Angelika May, 2005. "Measurement of aggregate risk with copulas," Econometrics Journal, Royal Economic Society, vol. 8(3), pages 428-454, December.
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    Cited by:

    1. Matthias R. Fengler & Helmut Herwartz & Christian Werner, 2012. "A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 10(3), pages 457-493, June.
    2. Xiangjin B. Chen & Param Silvapulle & Mervyn Silvapulle, 2013. "A Semiparametric Approach to Value-at-Risk, Expected Shortfall and Optimum Asset Allocation in Stock-Bond Portfolios," Monash Econometrics and Business Statistics Working Papers 14/13, Monash University, Department of Econometrics and Business Statistics.
    3. Fantazzini, Dean, 2011. "Analysis of multidimensional probability distributions with copula functions. II," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 23(3), pages 98-132.

    More about this item


    Association; Copula; Estimating Equation; Pseudolikelihood; Semiparametric.;

    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

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