Abstract
Although the variance-gamma distribution is a flexible model for log-returns of financial assets, so far it has found rather limited applications in finance and risk management. One of the reasons is that maximum likelihood estimation of its parameters is not straightforward. We develop an EM-type algorithm based on Nitithumbundit and Chan (An ECM algorithm for skewed multivariate variance gamma distribution in normal mean–variance representation, arXiv:1504.01239, 2015) that bypasses the evaluation of the full likelihood, which may be difficult because the density is not in closed form and is unbounded for small values of the shape parameter. Moreover, we study the relative efficiency of our approach with respect to the maximum likelihood estimation procedures implemented in the VarianceGamma and ghyp R packages. Extensive simulation experiments and real-data analyses suggest that the multicycle ECM algorithm gives the best results in terms of root-mean-squared-error, for both parameter and value-at-risk estimation. The performance of the routines in the ghyp R package is similar but not as good, whereas the VarianceGamma package produces worse results, especially when the shape parameter is small.
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We thank two anonymous referees whose valuable comments considerably improved an earlier version of the paper.
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An earlier version of this paper was presented at the 10th International Conference on Computational and Financial Econometrics (CFE 2016), Seville, December 9–11, 2016. We thank the participants for helpful comments and suggestions. The R codes used for estimating the VG distribution using the the MCECM, ECME and HECM algorithms and for computing the VaR are available at http://marcobee.weebly.com/software.html.
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Bee, M., Dickson, M.M. & Santi, F. Likelihood-based risk estimation for variance-gamma models. Stat Methods Appl 27, 69–89 (2018). https://doi.org/10.1007/s10260-017-0393-z
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DOI: https://doi.org/10.1007/s10260-017-0393-z