- Author
- Year
- 2013
- Title
- Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors
- Number of pages
- 17
- Publisher
- Amsterdam / Rotterdam: Tinbergen Institute
- Series
- Tinbergen Institute Discussion Paper, TI 2013-001/III
- Document type
- Working paper
- Faculty
- Faculty of Economics and Business (FEB)
- Institute
- Amsterdam School of Economics Research Institute (ASE-RI)
- 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 finite-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.
- Link
- Link
- Language
- English
- Persistent Identifier
- https://hdl.handle.net/11245/1.395016
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