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Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors

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  • Simon A. Broda

Abstract

Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closed-form expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic (MGHyp) random vectors. The MGHyp is a very flexible distribution which nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the first partial moment is also obtained, which plays a vital role in financial risk management. The proof involves a generalization of the classic inversion formula due to Gil-Pelaez (1951). Two applications are considered: first, the finite-sample distribution of the 2SLS estimator of a structural parameter. Second, the Value at Risk and Expected Shortfall of a quadratic portfolio with heavy-tailed risk factors.

Suggested Citation

  • Simon A. Broda, 2013. "Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors," UvA-Econometrics Working Papers 13-04, Universiteit van Amsterdam, Dept. of Econometrics.
  • Handle: RePEc:ame:wpaper:1304
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    References listed on IDEAS

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    1. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
    2. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
    3. Karim M. Abadir & Kaddour Hadri & Elias Tzavalis, 1999. "The Influence of VAR Dimensions on Estimator Biases," Econometrica, Econometric Society, vol. 67(1), pages 163-182, January.
    4. van GARDEREN, Kees Jan, 1997. "Exact geometry of explosive autoregressive models," CORE Discussion Papers 1997068, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Citations

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    Cited by:

    1. Simon A. Broda & Raymond Kan, 2016. "On distributions of ratios," Biometrika, Biometrika Trust, vol. 103(1), pages 205-218.
    2. Broda, Simon A. & Haas, Markus & Krause, Jochen & Paolella, Marc S. & Steude, Sven C., 2013. "Stable mixture GARCH models," Journal of Econometrics, Elsevier, vol. 172(2), pages 292-306.
    3. Arismendi, Juan C. & Broda, Simon, 2017. "Multivariate elliptical truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 29-44.

    More about this item

    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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