Broda, Simon A. and Arismendi Zambrano, Juan (2021) On quadratic forms in multivariate generalized hyperbolic random vectors. Biometrika, 108 (2). pp. 413-424. ISSN 1464-3510
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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 random vectors. This
exible distribution
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
numerical applications are considered: first, the finite-sample distribution of the two stage least
squares estimator of a structural parameter. Second, the Value at Risk and expected shortfall of a
quadratic portfolio with heavy-tailed risk factors. An empirical application is examined, in which
a portfolio of Dow Jones Industrial Index stock options is optimized with respect to its expected
shortfall. The results demonstrate the benefits of the analytical expression.
Item Type: | Article |
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Additional Information: | This is the preprint version of the published article, which is available at: Simon A Broda, Juan Arismendi Zambrano, On quadratic forms in multivariate generalized hyperbolic random vectors, Biometrika, Volume 108, Issue 2, June 2021, Pages 413–424, https://doi.org/10.1093/biomet/asaa067 |
Keywords: | Characteristic Function; Conditional Value at Risk; Expected Shortfall; Transform Inversion; Two Stage Least Squares; |
Academic Unit: | Faculty of Social Sciences > School of Business |
Item ID: | 15005 |
Identification Number: | 10.1093/biomet/asaa067 |
Depositing User: | Juan Arismendi Zambrano |
Date Deposited: | 11 Nov 2021 15:06 |
Journal or Publication Title: | Biometrika |
Publisher: | Oxford University Press |
Refereed: | Yes |
Related URLs: | |
URI: | https://mural.maynoothuniversity.ie/id/eprint/15005 |
Use Licence: | This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here |
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