Tail Probabilities and Partial Moments for Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors

Simon Broda

No 13-001/III, Tinbergen Institute Discussion Papers from Tinbergen Institute

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 exible distribution which nests, amongothers, 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 GilPelaez (1951).Two applications are considered: first, the nite-sample distribution of the 2SLS estimatorof a structural parameter. Second, the Value at Risk and Expected Shortfall of a quadraticportfolio with heavy-tailed risk factors.

Keywords: Finite Samples; Characteristic Function; Transform Inversion; 2SLS; CVaR; Expected Shortfall (search for similar items in EconPapers)
JEL-codes: C16 C36 C63 G11 G32 (search for similar items in EconPapers)
Date: 2013-01-08
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Citations: View citations in EconPapers (4)

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Related works:
Working Paper: Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors (2013)
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