Template-Type: ReDIF-Paper 1.0
Series: Tinbergen Institute Discussion Papers
Creation-Date: 2013-01-08
Number: 13-001/III
Author-Name: Simon A. Broda
Author-Workplace-Name: University of Amsterdam
Title: Tail Probabilities and Partial Moments for Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors
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 <I>t</I>, 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.
Classification-JEL: C16, C36, C63, G11, G32
Keywords: Finite Samples; Characteristic Function; Transform Inversion; 2SLS; CVaR; Expected Shortfall
File-Url: https://papers.tinbergen.nl/13001.pdf
File-Format: application/pdf
File-Size: 444020 bytes
Handle: RePEc:tin:wpaper:20130001