{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,31]],"date-time":"2026-10-31T03:54:43Z","timestamp":1793418883864,"version":"4.5.0"},"reference-count":54,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T00:00:00Z","timestamp":1598400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,5,15]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>This article presents exact and approximate expressions for tail probabilities and partial moments of quadratic forms in multivariate generalized hyperbolic random vectors. The derivations involve a generalization of the classic inversion formula for distribution functions (Gil-Pelaez, 1951). Two numerical applications are considered: the distribution of the two-stage least squares estimator and the expected shortfall of a quadratic portfolio.<\/jats:p>","DOI":"10.1093\/biomet\/asaa067","type":"journal-article","created":{"date-parts":[[2020,8,19]],"date-time":"2020-08-19T19:25:38Z","timestamp":1597865138000},"page":"413-424","source":"Crossref","is-referenced-by-count":5,"title":["On quadratic forms in multivariate generalized hyperbolic random vectors"],"prefix":"10.1093","volume":"108","author":[{"given":"Simon A","family":"Broda","sequence":"first","affiliation":[{"name":"Institute of Financial Services IFZ, Lucerne University of Applied Sciences, Suurstoffi 1, 6343 Rotkreuz, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juan Arismendi","family":"Zambrano","sequence":"additional","affiliation":[{"name":"Department of Economics, Finance and Accounting, National University of Ireland, Maynooth, County Kildare, Ireland juancarlos.arismendizambrano@mu.ie"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,8,26]]},"reference":[{"key":"2021051519490529500_B1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1111\/1468-0300.00091","article-title":"Expected shortfall: A natural coherent alternative to value at risk","volume":"31","author":"Acerbi,","year":"2002","journal-title":"Econ. Not."},{"key":"2021051519490529500_B2","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1016\/S0378-4266(02)00283-2","article-title":"On the coherence of expected shortfall","volume":"26","author":"Acerbi,","year":"2002","journal-title":"J. Banking Finan."},{"key":"2021051519490529500_B3","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1016\/j.jbankfin.2005.04.012","article-title":"Minimizing CVaR and VaR for a portfolio of derivatives","volume":"30","author":"Alexander,","year":"2006","journal-title":"J. Banking Finan."},{"key":"2021051519490529500_B4","doi-asserted-by":"crossref","first-page":"683","DOI":"10.2307\/1914090","article-title":"Distributions of estimates of coefficients of a single equation in a simultaneous system and their asymptotic expansions","volume":"41","author":"Anderson,","year":"1973","journal-title":"Econometrica"},{"key":"2021051519490529500_B5","first-page":"401","article-title":"Exponentially decreasing distributions for the logarithm of particle size","volume":"353","author":"Barndorff-Nielsen,","year":"1977","journal-title":"Proc. R. Soc. Lond."},{"key":"2021051519490529500_B6","volume-title":"Fundamental review of the trading book: A revised market risk framework","year":"2012"},{"key":"2021051519490529500_B7","doi-asserted-by":"crossref","first-page":"657","DOI":"10.2307\/2951662","article-title":"Alternative approximations to the distributions of instrumental variable estimators","volume":"62","author":"Bekker,","year":"1994","journal-title":"Econometrica"},{"key":"2021051519490529500_B8","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1086\/260062","article-title":"The pricing of options and corporate liabilities","volume":"81","author":"Black,","year":"1973","journal-title":"J. Polit. Econ."},{"key":"2021051519490529500_B9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/978-94-009-8549-0_3","volume-title":"Statistical Distributions in Scientific Work","author":"Bl\u00e6sild,","year":"1981"},{"key":"2021051519490529500_B10","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1002\/cpa.3160190403","article-title":"Uniform expansions of integrals with stationary points near algebraic singularity","volume":"19","author":"Bleistein,","year":"1966","journal-title":"Commun. Pure Appl. Math."},{"key":"2021051519490529500_B11","doi-asserted-by":"crossref","DOI":"10.1088\/1742-5468\/2010\/01\/P01005","article-title":"A generalized Fourier transform approach to risk measures","volume":"2010","author":"Bormetti,","year":"2010","journal-title":"J. Statist. Mech."},{"key":"2021051519490529500_B12","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1111\/j.1467-9965.2011.00482.x","article-title":"The expected shortfall of quadratic portfolios with heavy-tailed risk factors","volume":"22","author":"Broda,","year":"2012","journal-title":"Math. Finan."},{"key":"2021051519490529500_B13","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1093\/biomet\/asv052","article-title":"On distributions of ratios","volume":"103","author":"Broda,","year":"2016","journal-title":"Biometrika"},{"key":"2021051519490529500_B14","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.ecosta.2017.07.003","article-title":"Approximating expected shortfall for heavy-tailed distributions","volume":"8","author":"Broda,","year":"2018","journal-title":"Economet. Statist."},{"key":"2021051519490529500_B15","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1093\/jjfinec\/nbp011","article-title":"CHICAGO: A fast and accurate method for portfolio risk calculation","volume":"7","author":"Broda,","year":"2009","journal-title":"J. Finan. Economet."},{"key":"2021051519490529500_B16","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1002\/sim.4197","article-title":"Bias in causal estimates from Mendelian randomization studies with weak instruments","volume":"30","author":"Burgess,","year":"2011","journal-title":"Statist. Med."},{"key":"2021051519490529500_B17","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1093\/ije\/dyr036","article-title":"Avoiding bias from weak instruments in Mendelian randomization studies","volume":"40","author":"Burgess,","year":"2011","journal-title":"Int. J. Epidemiol."},{"key":"2021051519490529500_B18","doi-asserted-by":"crossref","first-page":"2712","DOI":"10.1214\/009053604000000689","article-title":"Saddlepoint approximation for moment generating functions of truncated random variables","volume":"32","author":"Butler,","year":"2004","journal-title":"Ann. Statist."},{"key":"2021051519490529500_B19","first-page":"91","article-title":"Evaluation of the two-stage least squares distribution function by Imhof\u2019s procedure","volume":"5","author":"Cribbett,","year":"1989","journal-title":"J. Quant. Econ."},{"key":"2021051519490529500_B20","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1002\/sim.6358","article-title":"The many weak instruments problem and Mendelian randomization","volume":"34","author":"Davies,","year":"2015","journal-title":"Statist. Med."},{"key":"2021051519490529500_B21","first-page":"427","article-title":"Distribution of the estimators for autoregressive time series with a unit root","volume":"74","author":"Dickey,","year":"1979","journal-title":"J. Am. Statist. Assoc."},{"key":"2021051519490529500_B22","first-page":"409","article-title":"Testing for serial correlation in least squares regression, I","volume":"37","author":"Durbin,","year":"1950","journal-title":"Biometrika"},{"key":"2021051519490529500_B23","doi-asserted-by":"crossref","first-page":"281","DOI":"10.2307\/3318481","article-title":"Hyperbolic distributions in finance","volume":"1","author":"Eberlein,","year":"1995","journal-title":"Bernoulli"},{"key":"2021051519490529500_B24","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1137\/110830319","article-title":"Inverting analytic characteristic functions and financial applications","volume":"4","author":"Feng,","year":"2013","journal-title":"SIAM J. Finan. Math."},{"key":"2021051519490529500_B25","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1017\/S0266466602184015","article-title":"The exact cumulative distribution function of a ratio of quadratic forms in normal variables, with application to the AR(1) model","volume":"18","author":"Forchini,","year":"2002","journal-title":"Economet. Theory"},{"key":"2021051519490529500_B26","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1017\/S0266466606060427","article-title":"On the bimodality of the exact distribution of the TSLS estimator","volume":"22","author":"Forchini,","year":"2006","journal-title":"Economet. Theory"},{"key":"2021051519490529500_B27","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.econlet.2006.09.022","article-title":"The exact distribution of the TSLS estimator for a non-Gaussian just-identified linear structural equation","volume":"95","author":"Forchini,","year":"2007","journal-title":"Econ. Lett."},{"key":"2021051519490529500_B28","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1093\/biomet\/38.3-4.481","article-title":"Note on the inversion theorem","volume":"38","author":"Gil-Pelaez,","year":"1951","journal-title":"Biometrika"},{"key":"2021051519490529500_B29","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1111\/1467-9965.00141","article-title":"Portfolio value-at-risk with heavy-tailed risk factors","volume":"12","author":"Glasserman,","year":"2002","journal-title":"Math. Finan."},{"key":"2021051519490529500_B30","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1080\/14697680903280483","article-title":"Efficient and robust portfolio optimization in the multivariate generalized hyperbolic framework","volume":"11","author":"Hellmich,","year":"2011","journal-title":"Quant. Finan."},{"key":"2021051519490529500_B31","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1017\/S0266466606060415","article-title":"Yet more on the exact properties of IV estimators","volume":"22","author":"Hillier,","year":"2006","journal-title":"Economet. Theory"},{"key":"2021051519490529500_B32","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.2307\/1914016","article-title":"A saddlepoint approximation to the distribution of the $k$-class estimator of a coefficient in a simultaneous system","volume":"47","author":"Holly,","year":"1979","journal-title":"Econometrica"},{"key":"2021051519490529500_B33","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1093\/biomet\/48.3-4.419","article-title":"Computing the distribution of quadratic forms in normal variables","volume":"48","author":"Imhof,","year":"1961","journal-title":"Biometrika"},{"key":"2021051519490529500_B34","doi-asserted-by":"crossref","DOI":"10.2139\/ssrn.1400965","volume-title":"Computing VaR and AVaR in infinitely divisible distributions","author":"Kim,","year":"2009"},{"key":"2021051519490529500_B35","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1017\/S0266466600011385","article-title":"Non-normal errors and the distribution of OLS and 2SLS structural estimators","volume":"2","author":"Knight,","year":"1986","journal-title":"Economet. Theory"},{"key":"2021051519490529500_B36","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0304-4076(92)90104-Y","article-title":"Testing the null hypothesis of stationarity against the alternative of a unit root","volume":"54","author":"Kwiatkowski,","year":"1992","journal-title":"J. Economet."},{"key":"2021051519490529500_B37","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1002\/sim.3034","article-title":"Mendelian randomization: Using genes as instruments for making causal inferences in epidemiology","volume":"27","author":"Lawlor,","year":"2008","journal-title":"Statist. Med."},{"key":"2021051519490529500_B38","doi-asserted-by":"crossref","first-page":"475","DOI":"10.2307\/1426607","article-title":"Saddlepoint approximations for the distribution of sums of independent random variables","volume":"12","author":"Lugannani,","year":"1980","journal-title":"Adv. Appl. Prob."},{"key":"2021051519490529500_B39","doi-asserted-by":"crossref","first-page":"181","DOI":"10.2307\/2951683","article-title":"On the exact small sample distribution of the instrumental variable estimator","volume":"60","author":"Maddala,","year":"1992","journal-title":"Econometrica"},{"key":"2021051519490529500_B40","first-page":"93","article-title":"The saddlepoint method and portfolio optionalities","volume":"19","author":"Martin,","year":"2006","journal-title":"Risk Mag."},{"key":"2021051519490529500_B41","doi-asserted-by":"crossref","first-page":"S125","DOI":"10.1086\/296497","article-title":"The distribution of the instrumental variables estimator and its $t$-ratio when the instrument is a poor one","volume":"63","author":"Nelson,","year":"1990","journal-title":"J. Bus."},{"key":"2021051519490529500_B42","doi-asserted-by":"crossref","first-page":"967","DOI":"10.2307\/2938359","article-title":"Some further results on the exact small sample properties of the instrumental variable estimator","volume":"58","author":"Nelson,","year":"1990","journal-title":"Econometrica"},{"key":"2021051519490529500_B43","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1017\/S0266466606060439","article-title":"A remark on bimodality and weak instrumentation in structural equation estimation","volume":"22","author":"Phillips,","year":"2006","journal-title":"Economet. Theory"},{"key":"2021051519490529500_B44","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s10959-010-0276-9","article-title":"Positive-part moments via the Fourier-Laplace transform","volume":"24","author":"Pinelis,","year":"2010","journal-title":"J. Theor. Prob."},{"key":"2021051519490529500_B45","doi-asserted-by":"crossref","first-page":"1971","DOI":"10.1002\/j.1538-7305.1968.tb01099.x","article-title":"Uniform asymptotic expansions for saddle point integrals \u2013 applications to a probability distribution occuring in noise theory","volume":"47","author":"Rice,","year":"1968","journal-title":"Bell Syst. Tech. J."},{"key":"2021051519490529500_B46","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.1080\/01621459.1968.10480921","article-title":"The exact distribution of a structural coefficient estimator","volume":"63","author":"Richardson,","year":"1968","journal-title":"J. Am. Statist. Assoc."},{"key":"2021051519490529500_B47","doi-asserted-by":"crossref","first-page":"21","DOI":"10.21314\/JOR.2000.038","article-title":"Optimization of conditional value at risk","volume":"2","author":"Rockafellar,","year":"2000","journal-title":"J. Risk"},{"key":"2021051519490529500_B48","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/BF02591508","article-title":"On the asymptotic distribution of sums of independent identically distributed random variables","volume":"4","author":"Ros\u00e9n,","year":"1961","journal-title":"Ark. Mat."},{"key":"2021051519490529500_B49","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1080\/01621459.1969.10501024","article-title":"The exact sampling distributions of ordinary least squares and two stage least squares estimates","volume":"64","author":"Sawa,","year":"1969","journal-title":"J. Am. Statist. Assoc."},{"key":"2021051519490529500_B50","doi-asserted-by":"crossref","first-page":"557","DOI":"10.2307\/2171753","article-title":"Instrumental variables regression with weak instruments","volume":"65","author":"Staiger,","year":"1997","journal-title":"Econometrica"},{"key":"2021051519490529500_B51","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1093\/ije\/dyh048","article-title":"Commentary: The concept of \u2018Mendelian randomization\u2019","volume":"33","author":"Thomas,","year":"2004","journal-title":"Int. J. Epidemiol."},{"key":"2021051519490529500_B52","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1214\/aoms\/1177705164","article-title":"The nonabsolute convergence of Gil-Pelaez\u2019 inversion integral","volume":"32","author":"Wendel,","year":"1961","journal-title":"Ann. Math. Statist."},{"key":"2021051519490529500_B53","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1111\/1468-0262.00247","article-title":"More results on the exact small sample properties of the instrumental variable estimator","volume":"69","author":"Woglom,","year":"2001","journal-title":"Econometrica"},{"key":"2021051519490529500_B54","doi-asserted-by":"crossref","first-page":"33","DOI":"10.3905\/jod.2010.17.3.033","article-title":"Analytical VaR and expected shortfall for quadratic portfolios","volume":"17","author":"Yueh,","year":"2010","journal-title":"J. Deriv."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asaa067\/34045864\/asaa067.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/108\/2\/413\/37938906\/asaa067.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/108\/2\/413\/37938906\/asaa067.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,15]],"date-time":"2021-05-15T19:49:51Z","timestamp":1621108191000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/108\/2\/413\/5897407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,26]]},"references-count":54,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,8,26]]},"published-print":{"date-parts":[[2021,5,15]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaa067","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2021,6,1]]},"published":{"date-parts":[[2020,8,26]]}}}