Download This PaperTail Probabilities and Partial Moments for Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors
Tinbergen Institute Discussion Paper 13-001/III
This article has been accepted for publication in Biometrika published by Oxford University Press. Cite as "Broda, Simon A. and Arismendi Zambrano, J (2020). On Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors. Biometrika, forthcoming. DOI: 10.1093/biomet/asaa067"
20 Pages Posted: 11 Jan 2013 Last revised: 19 Aug 2020
Date Written: January 8, 2013
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.
Keywords: Finite Samples, Characteristic Function, Transform Inversion, 2SLS, CVaR, Expected Shortfall
JEL Classification: C16, C36, C63, G11, G32
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