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| Type | Working Paper |
| Scope | Discipline-based scholarship |
| Title | Partial Moments for Quadratic Forms in Non-Gaussian Random Vectors: A Parametric Approach |
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| Series Name | SSRN |
| Number | 3369208 |
| ISSN | 1556-5068 |
| Date | 2019 |
| Abstract Text | 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 numerical 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. An empirical application is examined, where a portfolio of of Dow Jones Industrial Index (DJIA) stock options is optimised by minimising the expected shortfall. The empirical results show the benefits of the analytical expression. |
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| Digital Object Identifier | 10.2139/ssrn.3369208 |
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