Heavy tails and copulas topics in dependence modelling in economics and finance

9644_9789814689793_tp.indd 24/1/17 8:43 AM b2530   International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 01-Sep-16 11:03:06 AM 9644_9789814689793_tp.indd 24/1/17 8:43 AM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library HEAVY  TAILS  A ND  COPULAS Topics in Dependence Modelling in Economics and Finance Copyright © 2017 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4689-79-3 Desk Editor: Jiang Yulin Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore Yulin - Heavy Tails and Copulas.indd 10-11-16 9:01:33 AM January 24, 2017 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm To the memory of Kamil and grandmother Masguda R.I To A2 + I A.P v page v b2530   International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 01-Sep-16 11:03:06 AM January 24, 2017 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm Preface The idea of putting together a book on copulas and heavy tails has been brewing in our conversations for several years Both of us have been working on various problems in this field and we felt a monograph covering some of these results could have value There is a number of excellent and comprehensive treatments of copulas or heavy tails, with a statistical, mathematical, and risk management perspective This book is different in that it provides a unified approach to handling both copulas and heavy tails, and it takes an economics and finance perspective We are thinking of a diverse readership for this book First, it is academic and business readers, practitioners and theoreticians, who work with copula models and heavy tailed data The benefit here is to have various results under one title as opposed to scattered across academic journals and to outline leads for promising research directions and useful applications Second, it is graduate and advanced undergraduate students especially in econometrics and finance, but also in statistics, risk management and actuarial sciences, who look for a deeper understanding of dependence and heavy tails for their theses and degrees The level of mathematical rigor is that of a research paper but we tried to make the book readable for a PhD or Master’s student, with some parts suitable for senior undergraduate and honors students This book is based on recent and on-going research by the authors and their coauthors Specifically, Chapter draws on Ibragimov vii page vii January 24, 2017 viii 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm Heavy Tails and Copulas (2009b) Some of the results reviewed therein are also presented in Section 2.1.1 of the recent book by Ibragimov et al (2015) that deals with the analysis of models in economics and finance to heavytailedness Chapter draws on de la Pe˜ na et al (2006); de la Pe˜ na et al (2004); Medovikov and Prokhorov (2016) Chapter is based on Ibragimov and Walden (2011); and Ibragimov and Prokhorov (2016) Chapter is based on Prokhorov and Schmidt (2009); Burda and Prokhorov (2014) and Hill and Prokhorov (2016) Chapter draws on Prokhorov and Schmidt (2009); Huang and Prokhorov (2014); and Prokhorov et al (2015) These are fairly recent papers and the topics can be viewed as part of the state-of-the-art in the area More importantly, this book is not equal to the sum of the papers The reasons are that, first, we not use entire papers as chapters — many proofs are omitted and some technical details are dropped, targeting a wider audience and assuming that interested readers will look them up in the original Second, we provide a leitmotif for each chapter that shows how we think the chapters are linked into a logical and readable sequence The ultimate goal is to provide a framework for thinking about fat tails and copulas in economics and finance, rather than to review the content of the papers R M Ibragimov, London, 2016 A B Prokhorov, Sydney, 2016 page viii January 24, 2017 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm Foreword The copula is a generally applicable and flexible tool for handling multivariate non-Gaussian dependence Sklar’s theorem implies that a multivariate distribution function can be written as the composition of the copula function with univariate cumulative distribution functions as arguments Hence for multivariate modelling, one can separate the modelling of univariate margins from the dependence structure as summarized by the copula This is especially useful if univariate margins have heavy tails and/or joint tail probabilities have more dependence than Gaussian dependence In this book, probabilistic properties are studied on the effect of heavy-tailedness and joint tail dependence on risk measures such as Value-at-Risk, and these properties have relevance to portfolio diversification Theory and tools are presented so that under some dependence assumptions, bounds on such quantities as option prices can be obtained, and the effect of the strength of dependence can be studied In practical data analysis using copulas, any parametric model being used is misspecified to some extent Without a physical or stochastic basis, “true” multivariate distributions cannot be expected to have simple forms, but flexible parametric constructions, such as vine and factor models for dependence, might provide good approximations Generally, a copula model might be satisfactory if it has relevant dependence and univariate/joint tail properties, or matches some “generalized” moments The latter chapters of this book have ix page ix January 24, 2017 x 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm Heavy Tails and Copulas results on estimation procedures that might be robust to a small degree of model misspecification This book differentiates itself from other books with “copula” in the title with its viewpoint via economic theory I commend the authors for writing this book and bringing together useful research in heavy tails and copula dependence, with orientation to economics and finance It should help to stimulate further research on the theme, and I look forward to seeing future developments Harry Joe 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expenditures, Health Economics 21, 12, pp 1444–1455 Wooldridge, J (1991) Specification testing and quasi-maximum likelihood estimation, Journal of Econometrics 48, pp 29–55 Zastavnyi, V P (1993) Positive definite functions depending on the norm, Russian Journal of Mathematical Physics 1, pp 511–522 Zheng, Y (2011) Shape restriction of the multi-dimensional Bernstein prior for density functions, Statistics and Probability Letters 81, 6, pp 647–651 Zheng, Y., Zhu, J., and Roy, A (2010) Nonparametric Bayesian inference for the spectral density function of a random field, Biometrika 97, pp 238–245 Zimmer, D (2012) The role of copulas in the housing crisis, Review of Economics and Statistics 94, 2, pp 607–620 page 278 January 24, 2017 13:38 Heavy Tails and Copulas Bibliography 9in x 6in b2639-bib page 279 279 Zimmer, D M and Trivedi, P K (2006) Using trivariate copulas to model sample selection and treatment effects: Application to family health care demand, Journal of Business and Economic Statistics 24, pp 63–76 Zolotarev, V (1991) Reflection on the classical theory of limit theorems, Theory of Probability and Its Applications 36, pp 124–137 Zolotarev, V M (1986) One-Dimensional Stable Distributions (American Mathematical Society, Providence, RI) b2530   International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 01-Sep-16 11:03:06 AM January 24, 2017 13:38 Heavy Tails and Copulas 9in x 6in b2639-index Index α-symmetric distributions, 113, 116, 127–130, 152, 153, 156 adaptive sparsity, 260 Ali-Mikhail-Haq (AMH) copulas, 120 Asian options, 97, 102, 103 complete decoupling inequalities, 81, 87, 91, 92 conditional moment test of copula robustness, 229, 231, 235 conditionally symmetric martingale difference, 73, 74, 112 consistency, 176, 182, 183, 216 Bayesian posterior consistency, 215, 216 Continuously Updated Estimator (CUE), 217, 219 convolutions of distributions, 113, 115, 125, 127, 129, 152, 156 convolutions of stables (CS), 114, 127 copula, 4–9, 14–17 t-copula, absolutely continuous copula, 4, copula density, definition, 4, 5, 10 Frechet-Hoeffding bounds, Gaussian copula, 6, 11, 12, 16 Sklar theorem, copula robustness, 209 copula validity, 200, 201, 220 copula validity test, 231 copulas with cubic sections, 118, 124 curse of dimensionality, 259 “blanket” copula goodness-of-fit test, 230, 231, 246 Bernstein copula, 176, 207, 210, 213, 221 Brownian motion with non-Gaussian marginals, 67 Canonical MLE (CMLE), 217 Cauchy distribution, 10 coherent risk measure, 22, 34 convexity, 34 monotonicity, 26 positive homogeneity, 26 subadditivity, 26, 35 translation invariance, 26 common shocks, 116, 125–128, 130–132, 134, 137–140, 142, 143, 147, 150, 152–156 additive common shocks, 130, 132, 156 multiplicative common shocks, 126, 150 complete characterizations of distributions, 49, 57, 104 Dantzig selector, 211, 212 Dirichlet process prior, 213 distribution, 2–4, 6–9, 11 281 page 281 January 24, 2017 282 13:38 Heavy Tails and Copulas 9in x 6in b2639-index Heavy Tails and Copulas Cauchy, L´evy, Pareto, 10 cdf, 3–8 joint distribution, 4, k-dimensional marginal, marginal distribution, 4, multivariate cdf, one-dimensional cdf, Student-t, 6, 7, 10 diversification of portfolio risk, 26 efficiency, 172, 175, 180, 181, 184, 185, 191, 193, 194, 204, 206, 207, 209, 211, 212, 217, 220, 221 Bayesian efficiency, 207, 213 relative efficiency, 146, 150, 209 semiparametric efficiency bound, 181, 210, 211, 220 eigenspectrum of information matrix, 229–232, 235–237, 240, 241, 243, 249, 250 entropy generalized entropy, 86, 87 relative entropy, 82, 84–86, 91, 104 Euclidean Empirical Likelihood, 219 European options, 98 Expected Shortfall (ES), 39, 114 robust expected shortfall estimation, 219 Eyraud-Farlie-Gumbel-Morgenstern (EFGM) copulas, 54, 79, 120 factor-copulas, 260 financial contagion, financial crisis 1982 crisis, 15 1987 crisis, 15 1997 crisis, 15 2008 crisis, 15 Black Monday, 15 Fourier copulas, 62, 79, 80 Frank copulas, 120, 121 Full Maximum Likelihood Estimator (FMLE), 175, 188, 209 GEL Estimator with Imbedded Tail-Trimming (GELITT), 219 Generalized Empirical Likelihood Estimation (GEL), 187, 217, 219 GARCH models, 217 Generalized Information Matrix Tests (GIMT), 241, 248, 250 Generalized Method of Moments (GMM), 175, 184 googol, heavy (fat) tailed data, heavy tailed distributions extremely heavy tailed distributions, 19–21, 25, 30, 34, 37, 38 moderately heavy tailed distributions, 20, 21, 25, 29, 38 Hoeffding’s Φ2 , 82, 84, 93 vector Hoeffding’s Φ2 , 82, 84, 93 Inference Function for Margins (IFM), 173–175, 185 Improved PMLE (IPMLE), 205, 206 Improved QMLE (IQMLE), 185, 187, 220 independent r.v.’s, 48, 52, 56, 58, 59, 61, 69, 75, 87, 91 information matrix equivalence, 235–237 information matrix test, 237, 240, 241, 243, 248, 250 Kendall’s τ , 82–84 Kimeldorf and Sampson copulas, 124 l1 -norm shrinkage, 212 lacunary trigonometric system, 58, 59 large deviations, 10 limits of diversification, 37, 113, 116, 124, 155, 156 Levy distribution, 10 log-concave distributions (LC), 22–25, 38, 130, 133, 152 Lorenz dominance, 26 page 282 January 24, 2017 13:38 Heavy Tails and Copulas 9in x 6in b2639-index Index m-dependent r.v.’s, 58 majorization of vectors, 26 marginally symmetric random variable, 202 Markov process, 48, 60–62, 65–72, 74–80, 109–112 martingale-difference sequence, 59, 98, 100, 103 multiplicative system, 58–60, 75, 104 multivariate divergence, 82, 85–87, 90, 104 orthogonal r.v.’s, 57, 58, 105 overidentifying restrictions test, 232 p-majorization of vectors, 26 Pareto distribution, 10 Pearson’s φ2 , 82, 83, 85, 91 Penalized MLE, 211 Plackett copulas, 120, 121 polynomial copulas, 118 power copulas, 55, 61, 76 power law family of distributions, extremely heavy tails, 9, 14–17 moderately heavy tails, 15 moments of power law family, tail index, power-type copulas, 118, 123, 124 Pseudo-Maximum Likelihood Estimator (PMLE), 220 Quasi-Maximum Likelihood Estimator (QMLE), 208 r-independent r.v.’s, 59 radially symmetric copula, 202, 204 radially symmetric random variables, 201, 202, 204 random effects estimator, 146 reduction property, 60, 69, 74 redundant copula, 209 regularly varying heavy tails, 35 return on financial index, 131, 135, 136, 140, 141, 151 Global Dow, 139 283 Thomson Reuters Equal Weight Continuous Commodity Index, 139 Thomson Reuters/Jefferies CRB Index, 139 Value Line Arithmetic Index, 139 robustness, 14–17 robustness of diversification in Value-at-Risk models, 14 robustness of methods to heavy tails, 15 robustness of methods to copula misspecification, 15 robustness of models to heavy tails, 14 Sarmanov copulas, 121 scaled squared Hellinger distance, 86 Schur-concave functions, 21, 133 Schur-convex functions, 28, 133 sequential copula constructions, 260 Shannon (Kullback-Leibler) mutual information, 85 Sieve Maximum Likelihood Estimator (SMLE), 182, 209–212 size distortions of information matrix test, 237, 240, 241, 246, 248, 250 slowly varying at infinity function, 35 sparsity, 212 Spearman’s ρ, 82–84 spherical distributions, 125–127, 130, 156 stable distributions, 10–14 definition, index of stability, 10, 11 strictly stable, 11 star-product of copulas, 60, 61 strictly stationary r.v.’s, 57 tail dependence, 2, 3, 16 abnormal jumps and drops, asymmetric tail dependence, trinomial model of option pricing, 97–104 page 283 January 24, 2017 284 13:38 Heavy Tails and Copulas 9in x 6in b2639-index Heavy Tails and Copulas two-step test of copula robustness, 229, 230, 232 U -statistics, 47, 48, 54, 56, 61, 68–71, 88, 91, 105 unimodal distributions, 34, 42 upper and lower tail dependence, 85 upper and lower tail-order, 258 Value-at-Risk, 19 definition, 21, 25, 34 sharp bounds for, 20 subadditivity, 28, 34, 35 vine-copulas, 259 weakly stationary r.v.’s, 57 page 284 ... enquiries@stallionpress.com Printed in Singapore Yulin - Heavy Tails and Copulas. indd 10-11-16 9:01:33 AM January 24, 2017 13:39 Heavy Tails and Copulas 9in x 6in b2639-fm To the memory of Kamil and grandmother... Library Cataloguing -in- Publication Data A catalogue record for this book is available from the British Library HEAVY TAILS  A ND COPULAS Topics in Dependence Modelling in Economics and Finance Copyright... marginal cdf ’s FX1 , , FXd are all continuous, then the copula is unique and can January 24, 2017 13:38 Heavy Tails and Copulas 9in x 6in b2639-ch01 Heavy Tails and Copulas be obtained via inversion: