Dynamic Copulas For Finance

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Dynamic Copula Methods in Finance

Author : Umberto Cherubini
Publisher : John Wiley & Sons
ISBN 13 : 1119954525
Total Pages : 287 pages
Book Rating : 4.1/5 (199 download)

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Book Synopsis Dynamic Copula Methods in Finance by : Umberto Cherubini

Download or read book Dynamic Copula Methods in Finance written by Umberto Cherubini and published by John Wiley & Sons. This book was released on 2011-10-20 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: The latest tools and techniques for pricing and risk management This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications. The first part of the book will briefly introduce the standard the theory of copula functions, before examining the link between copulas and Markov processes. It will then introduce new techniques to design Markov processes that are suited to represent the dynamics of market risk factors and their co-movement, providing techniques to both estimate and simulate such dynamics. The second part of the book will show readers how to apply these methods to the evaluation of pricing of multivariate derivative contracts in the equity and credit markets. It will then move on to explore the applications of joint temporal and cross-section aggregation to the problem of risk integration.

Dynamic Copulas for Finance

Author : Valentin Braun
Publisher : BoD – Books on Demand
ISBN 13 : 3844100407
Total Pages : 178 pages
Book Rating : 4.8/5 (441 download)

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Book Synopsis Dynamic Copulas for Finance by : Valentin Braun

Download or read book Dynamic Copulas for Finance written by Valentin Braun and published by BoD – Books on Demand. This book was released on 2011 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interactions of financial securities are crucial to determine possible portfolio losses. Although this fact is well understood, two questions remain: What causes changes in the dependence structure of financial assets? How can fluctuating dependencies be measured? The most common approach to identify the amplitude of financial assets' interactions are linear correlation coefficients. However, they fail to comprise shifts in the dependence structure. Alternatively, Copulas are a more flexible dependence measurement. This book focuses on the development of Dynamic Copula frameworks by implementing stochastic parameters into Archimedian and Elliptical Copula functions. In contrast to static correlation measures, the Dynamic Copulas are able to replicate unstable financial market interactions. Various Dynamic Copulas are applied to global stock, bond, commodity and exchange rate data to calculate the correlation time paths, which explain financial market reactions to economic shocks. Furthermore, the interactions of dependencies, volatility and returns are analyzed, to determine the efficiency of portfolio diversification in regards to wealth protection. Portfolio risks are estimated through Dynamic Copulas to demonstrate their abilities to replicate financial market interactions accurately. Additionally, this analysis reveals the impact of changing dependence intensities on the magnitude of possible portfolio losses. Finally, the Dynamic Copulas are utilized to allocate higher moment optimal portfolios. This examination emphasizes the effect of inaccurate correlation estimates on the portfolio choice.

Credit Models and the Crisis

Author : Damiano Brigo
Publisher : John Wiley & Sons
ISBN 13 : 0470971436
Total Pages : 212 pages
Book Rating : 4.4/5 (79 download)

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Book Synopsis Credit Models and the Crisis by : Damiano Brigo

Download or read book Credit Models and the Crisis written by Damiano Brigo and published by John Wiley & Sons. This book was released on 2010-10-28 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The recent financial crisis has highlighted the need for better valuation models and risk management procedures, better understanding of structured products, and has called into question the actions of many financial institutions. It has become commonplace to blame the inadequacy of credit risk models, claiming that the crisis was due to sophisticated and obscure products being traded, but practitioners have for a long time been aware of the dangers and limitations of credit models. It would seem that a lack of understanding of these models is the root cause of their failures but until now little analysis had been published on the subject and, when published, it had gained very limited attention. Credit Models and the Crisis is a succinct but technical analysis of the key aspects of the credit derivatives modeling problems, tracing the development (and flaws) of new quantitative methods for credit derivatives and CDOs up to and through the credit crisis. Responding to the immediate need for clarity in the market and academic research environments, this book follows the development of credit derivatives and CDOs at a technical level, analyzing the impact, strengths and weaknesses of methods ranging from the introduction of the Gaussian Copula model and the related implied correlations to the introduction of arbitrage-free dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. It also illustrates the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities, and explains why the Gaussian Copula model is still used in its base correlation formulation. The book reports both alarming pre-crisis research and market examples, as well as commentary through history, using data up to the end of 2009, making it an important addition to modern derivatives literature. With banks and regulators struggling to fully analyze at a technical level, many of the flaws in modern financial models, it will be indispensable for quantitative practitioners and academics who want to develop stable and functional models in the future.

Multivariate GARCH and Dynamic Copula Models for Financial Time Series

Author : Martin Grziska
Publisher : Pro BUSINESS
ISBN 13 : 3863868439
Total Pages : 191 pages
Book Rating : 4.8/5 (638 download)

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Book Synopsis Multivariate GARCH and Dynamic Copula Models for Financial Time Series by : Martin Grziska

Download or read book Multivariate GARCH and Dynamic Copula Models for Financial Time Series written by Martin Grziska and published by Pro BUSINESS. This book was released on 2015-02-05 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents several non-parametric and parametric models for estimating dynamic dependence between financial time series and evaluates their ability to precisely estimate risk measures. Furthermore, the different dependence models are used to analyze the integration of emerging markets into the world economy. In order to analyze numerous dependence structures and to discover possible asymmetries, two distinct model classes are investigated: the multivariate GARCH and Copula models. On the theoretical side a new dynamic dependence structure for multivariate Archimedean Copulas is introduced which lifts the prevailing restriction to two dimensions and extends the multivariate dynamic Archimedean Copulas to more than two dimensions. On this basis a new mixture copula is presented using the newly invented multivariate dynamic dependence structure for the Archimedean Copulas and mixing it with multivariate elliptical copulas. Simultaneously a new process for modeling the time-varying weights of the mixture copula is introduced: this specification makes it possible to estimate various dependence structures within a single model. The empirical analysis of different portfolios shows that all equity portfolios and the bond portfolios of the emerging markets exhibit negative asymmetries, i.e. increasing dependence during market downturns. However, the portfolio consisting of the developed market bonds does not show any negative asymmetries. Overall, the analysis of the risk measures reveals that parametric models display portfolio risk more precisely than non-parametric models. However, no single parametric model dominates all other models for all portfolios and risk measures. The investigation of dependence between equity and bond portfolios of developed countries, proprietary, and secondary emerging markets reveals that secondary emerging markets are less integrated into the world economy than proprietary. Thus, secondary emerging markets are moresuitable to diversify a portfolio consisting of developed equity or bond indices than proprietary.

Copulae and Multivariate Probability Distributions in Finance

Author : Alexandra Dias
Publisher : Routledge
ISBN 13 : 1317976908
Total Pages : 310 pages
Book Rating : 4.3/5 (179 download)

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Book Synopsis Copulae and Multivariate Probability Distributions in Finance by : Alexandra Dias

Download or read book Copulae and Multivariate Probability Distributions in Finance written by Alexandra Dias and published by Routledge. This book was released on 2013-08-21 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Portfolio theory and much of asset pricing, as well as many empirical applications, depend on the use of multivariate probability distributions to describe asset returns. Traditionally, this has meant the multivariate normal (or Gaussian) distribution. More recently, theoretical and empirical work in financial economics has employed the multivariate Student (and other) distributions which are members of the elliptically symmetric class. There is also a growing body of work which is based on skew-elliptical distributions. These probability models all exhibit the property that the marginal distributions differ only by location and scale parameters or are restrictive in other respects. Very often, such models are not supported by the empirical evidence that the marginal distributions of asset returns can differ markedly. Copula theory is a branch of statistics which provides powerful methods to overcome these shortcomings. This book provides a synthesis of the latest research in the area of copulae as applied to finance and related subjects such as insurance. Multivariate non-Gaussian dependence is a fact of life for many problems in financial econometrics. This book describes the state of the art in tools required to deal with these observed features of financial data. This book was originally published as a special issue of the European Journal of Finance.

Handbook of Financial Time Series

Author : Torben Gustav Andersen
Publisher : Springer Science & Business Media
ISBN 13 : 3540712976
Total Pages : 1045 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Handbook of Financial Time Series by : Torben Gustav Andersen

Download or read book Handbook of Financial Time Series written by Torben Gustav Andersen and published by Springer Science & Business Media. This book was released on 2009-04-21 with total page 1045 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook of Financial Time Series gives an up-to-date overview of the field and covers all relevant topics both from a statistical and an econometrical point of view. There are many fine contributions, and a preamble by Nobel Prize winner Robert F. Engle.

Modelling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas

Author : Valentin Braun
Publisher :
ISBN 13 :
Total Pages : 36 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Modelling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas by : Valentin Braun

Download or read book Modelling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas written by Valentin Braun and published by . This book was released on 2016 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose a multidimensional extension for Patton's (2006) bivariate Dynamic Copulas. We also introduce a Dynamic Mixture Copula whose parameters and weights follow well defined dynamic processes. Both approaches are more flexible to adapt to financial data than currently available Copula models. We utilize the G7 stocks and bonds data to demonstrate the advantages of the proposed Dynamic Copulas. The object of interest is the analysis of the characteristics of financial market interactions. We apply the proposed dynamic models to demonstrate that neither stock nor bond market interactions are time-stable. Further, we conduct analyses to demonstrate that our suggested Dynamic Copulas are flexible enough to capture time-instable correlation patterns and to account for tail dependencies. Finally, we quantify the interaction characteristics of the G7 stocks and bonds markets and find that stocks tend to drop simultaneously during market turmoil. In contrast, bond markets offer diversification effects that tend to increase during market turbulences.

Dependence Modeling

Author : Harry Joe
Publisher : World Scientific
ISBN 13 : 981429988X
Total Pages : 370 pages
Book Rating : 4.8/5 (142 download)

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Book Synopsis Dependence Modeling by : Harry Joe

Download or read book Dependence Modeling written by Harry Joe and published by World Scientific. This book was released on 2011 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. Introduction : Dependence modeling / D. Kurowicka -- 2. Multivariate copulae / M. Fischer -- 3. Vines arise / R.M. Cooke, H. Joe and K. Aas -- 4. Sampling count variables with specified Pearson correlation : A comparison between a naive and a C-vine sampling approach / V. Erhardt and C. Czado -- 5. Micro correlations and tail dependence / R.M. Cooke, C. Kousky and H. Joe -- 6. The Copula information criterion and Its implications for the maximum pseudo-likelihood estimator / S. Gronneberg -- 7. Dependence comparisons of vine copulae with four or more variables / H. Joe -- 8. Tail dependence in vine copulae / H. Joe -- 9. Counting vines / O. Morales-Napoles -- 10. Regular vines : Generation algorithm and number of equivalence classes / H. Joe, R.M. Cooke and D. Kurowicka -- 11. Optimal truncation of vines / D. Kurowicka -- 12. Bayesian inference for D-vines : Estimation and model selection / C. Czado and A. Min -- 13. Analysis of Australian electricity loads using joint Bayesian inference of D-vines with autoregressive margins / C. Czado, F. Gartner and A. Min -- 14. Non-parametric Bayesian belief nets versus vines / A. Hanea -- 15. Modeling dependence between financial returns using pair-copula constructions / K. Aas and D. Berg -- 16. Dynamic D-vine model / A. Heinen and A. Valdesogo -- 17. Summary and future directions / D. Kurowicka

Modeling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas

Author : Valentin Braun
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Book Synopsis Modeling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas by : Valentin Braun

Download or read book Modeling Asymmetric Dependence of Financial Returns with Multivariate Dynamic Copulas written by Valentin Braun and published by . This book was released on 2011 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose a multidimensional extension for Patton's (2006) bi-variate Dynamic Copulas. We also introduce a Dynamic Mixture Copula whose parameters and weights follow well defined dynamic processes. Both approaches improve the Copulas' flexibilities and their adaptabilities to financial data. We utilize one G7 Stocks and one Multi-Asset Classes dataset to demonstrate the advantages of the proposed Dynamic Copulas. The object of interest are the accuracies of out-of-sample one-week portfolio Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) predictions and the time-instability of financial market correlations. From our empirical analyses we report three main findings. First, Dynamic Copulas which explicitly account for left tail dependence tremendously improve the portfolio VaR and CVaR forecasts for both datasets. Second, severe negative Stock market returns occur jointly with intensifying volatility levels and tightening dependencies what eliminates diversification effects when they are needed the most. Diversification among multiple asset classes significantly reduces the risk of correlation meltdowns and hence the probability of joint price drops. Third, the Euro introduction increased the correlations of the German, the Italian and the French Stock markets to about 0.9.

Elements of Copula Modeling with R

Author : Marius Hofert
Publisher : Springer
ISBN 13 : 3319896350
Total Pages : 267 pages
Book Rating : 4.3/5 (198 download)

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Book Synopsis Elements of Copula Modeling with R by : Marius Hofert

Download or read book Elements of Copula Modeling with R written by Marius Hofert and published by Springer. This book was released on 2019-01-09 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the main theoretical findings related to copulas and shows how statistical modeling of multivariate continuous distributions using copulas can be carried out in the R statistical environment with the package copula (among others). Copulas are multivariate distribution functions with standard uniform univariate margins. They are increasingly applied to modeling dependence among random variables in fields such as risk management, actuarial science, insurance, finance, engineering, hydrology, climatology, and meteorology, to name a few. In the spirit of the Use R! series, each chapter combines key theoretical definitions or results with illustrations in R. Aimed at statisticians, actuaries, risk managers, engineers and environmental scientists wanting to learn about the theory and practice of copula modeling using R without an overwhelming amount of mathematics, the book can also be used for teaching a course on copula modeling.

Copula Methods in Finance

Author : Umberto Cherubini
Publisher : John Wiley & Sons
ISBN 13 : 0470863455
Total Pages : 310 pages
Book Rating : 4.4/5 (78 download)

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Book Synopsis Copula Methods in Finance by : Umberto Cherubini

Download or read book Copula Methods in Finance written by Umberto Cherubini and published by John Wiley & Sons. This book was released on 2004-10-22 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Copula Methods in Finance is the first book to address the mathematics of copula functions illustrated with finance applications. It explains copulas by means of applications to major topics in derivative pricing and credit risk analysis. Examples include pricing of the main exotic derivatives (barrier, basket, rainbow options) as well as risk management issues. Particular focus is given to the pricing of asset-backed securities and basket credit derivative products and the evaluation of counterparty risk in derivative transactions.

Simulating Copulas

Author : Jan-Frederik Mai
Publisher : World Scientific
ISBN 13 : 1848168748
Total Pages : 310 pages
Book Rating : 4.8/5 (481 download)

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Book Synopsis Simulating Copulas by : Jan-Frederik Mai

Download or read book Simulating Copulas written by Jan-Frederik Mai and published by World Scientific. This book was released on 2012 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the reader with a background on simulating copulas and multivariate distributions in general. It unifies the scattered literature on the simulation of various families of copulas (elliptical, Archimedean, Marshall-Olkin type, etc.) as well as on different construction principles (factor models, pair-copula construction, etc.). The book is self-contained and unified in presentation and can be used as a textbook for advanced undergraduate or graduate students with a firm background in stochastics. Alongside the theoretical foundation, ready-to-implement algorithms and many examples make this book a valuable tool for anyone who is applying the methodology.Errata(s)Errata (128 KB)

Dynamic Models for Volatility and Heavy Tails

Author : Andrew C. Harvey
Publisher : Cambridge University Press
ISBN 13 : 1107328780
Total Pages : 281 pages
Book Rating : 4.1/5 (73 download)

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Book Synopsis Dynamic Models for Volatility and Heavy Tails by : Andrew C. Harvey

Download or read book Dynamic Models for Volatility and Heavy Tails written by Andrew C. Harvey and published by Cambridge University Press. This book was released on 2013-04-22 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volatility of financial returns changes over time and, for the last thirty years, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models have provided the principal means of analyzing, modeling and monitoring such changes. Taking into account that financial returns typically exhibit heavy tails - that is, extreme values can occur from time to time - Andrew Harvey's new book shows how a small but radical change in the way GARCH models are formulated leads to a resolution of many of the theoretical problems inherent in the statistical theory. The approach can also be applied to other aspects of volatility. The more general class of Dynamic Conditional Score models extends to robust modeling of outliers in the levels of time series and to the treatment of time-varying relationships. The statistical theory draws on basic principles of maximum likelihood estimation and, by doing so, leads to an elegant and unified treatment of nonlinear time-series modeling.

Asymmetric Dependence in Finance

Author : Jamie Alcock
Publisher : John Wiley & Sons
ISBN 13 : 1119289017
Total Pages : 312 pages
Book Rating : 4.1/5 (192 download)

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Book Synopsis Asymmetric Dependence in Finance by : Jamie Alcock

Download or read book Asymmetric Dependence in Finance written by Jamie Alcock and published by John Wiley & Sons. This book was released on 2018-06-05 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Avoid downturn vulnerability by managing correlation dependency Asymmetric Dependence in Finance examines the risks and benefits of asset correlation, and provides effective strategies for more profitable portfolio management. Beginning with a thorough explanation of the extent and nature of asymmetric dependence in the financial markets, this book delves into the practical measures fund managers and investors can implement to boost fund performance. From managing asymmetric dependence using Copulas, to mitigating asymmetric dependence risk in real estate, credit and CTA markets, the discussion presents a coherent survey of the state-of-the-art tools available for measuring and managing this difficult but critical issue. Many funds suffered significant losses during recent downturns, despite having a seemingly well-diversified portfolio. Empirical evidence shows that the relation between assets is much richer than previously thought, and correlation between returns is dependent on the state of the market; this book explains this asymmetric dependence and provides authoritative guidance on mitigating the risks. Examine an options-based approach to limiting your portfolio's downside risk Manage asymmetric dependence in larger portfolios and alternate asset classes Get up to speed on alternative portfolio performance management methods Improve fund performance by applying appropriate models and quantitative techniques Correlations between assets increase markedly during market downturns, leading to diversification failure at the very moment it is needed most. The 2008 Global Financial Crisis and the 2006 hedge-fund crisis provide vivid examples, and many investors still bear the scars of heavy losses from their well-managed, well-diversified portfolios. Asymmetric Dependence in Finance shows you what went wrong, and how it can be corrected and managed before the next big threat using the latest methods and models from leading research in quantitative finance.

Credit Correlation

Author : Alexander Lipton
Publisher : World Scientific
ISBN 13 : 9812709509
Total Pages : 178 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis Credit Correlation by : Alexander Lipton

Download or read book Credit Correlation written by Alexander Lipton and published by World Scientific. This book was released on 2008 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: The recent growth of credit derivatives has been explosive. The global credit derivatives market grew in notional value from $1 trillion to $20 trillion from 2000 to 2006. However, understanding the true nature of these instruments still poses both theoretical and practical challenges. For a long time now, the framework of Gaussian copulas parameterized by correlation, and more recently base correlation, has provided an adequate, if unintuitive, description of the market. However, the increased liquidity in credit indices and index tranches, as well as the proliferation of exotic instruments such as forward starting tranches, options on tranches, leveraged super senior tranches, and the like, have made it imperative to come up with models that describe market reality better. This book, originally and concurrently published in the International Journal of Theoretical and Applied Finance, Vol. 10, No. 4, 2007, agrees that base correlation has outlived its usefulness; opinions of how to replace it, however, are divided. Both the top-down and bottom-up approaches for describing the dynamics of credit baskets are presented, and pro and contra arguments are put forward. Readers will decide which direction is the most promising one at the moment. However, it is hoped that, in the near future, models that transcend base correlation will be proposed and accepted by the market. Sample Chapter(s). Introduction (31 KB). Chapter 1: L(r)vy Simples Tructural Models (209 KB). Contents: L(r)vy Simple Structural Models (M Baxter); Cluster-Based Extension of the Generalized Poisson Loss Dynamics and Consistency with Single Names (D Brigo et al.); Stochastic Intensity Modeling for Structured Credit Exotics (A Chaposvsky et al.); Large Portfolio Credit Risk Modeling (M H A Davis & J C Esparragoza-Rodriguez); Empirical Copulas for CDO Tranche Pricing Using Relative Entropy (M A H Dempster et al.); Pricing and Hedging in a Dynamic Credit Model (Y Elouerkhaoui); Joint Distributions of Portfolio Losses and Exotic Portfolio Products (F Epple et al.); On the Term Structure of Loss Distributions: A Forward Model Approach (J Sidenius). Readership: Professionals, academics and students in the areas of finance and bank

Recent Econometric Techniques for Macroeconomic and Financial Data

Author : Gilles Dufrénot
Publisher : Springer Nature
ISBN 13 : 3030542521
Total Pages : 387 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Recent Econometric Techniques for Macroeconomic and Financial Data by : Gilles Dufrénot

Download or read book Recent Econometric Techniques for Macroeconomic and Financial Data written by Gilles Dufrénot and published by Springer Nature. This book was released on 2020-11-21 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a comprehensive overview of the latest econometric methods for studying the dynamics of macroeconomic and financial time series. It examines alternative methodological approaches and concepts, including quantile spectra and co-spectra, and explores topics such as non-linear and non-stationary behavior, stochastic volatility models, and the econometrics of commodity markets and globalization. Furthermore, it demonstrates the application of recent techniques in various fields: in the frequency domain, in the analysis of persistent dynamics, in the estimation of state space models and new classes of volatility models. The book is divided into two parts: The first part applies econometrics to the field of macroeconomics, discussing trend/cycle decomposition, growth analysis, monetary policy and international trade. The second part applies econometrics to a wide range of topics in financial economics, including price dynamics in equity, commodity and foreign exchange markets and portfolio analysis. The book is essential reading for scholars, students, and practitioners in government and financial institutions interested in applying recent econometric time series methods to financial and economic data.

An Introduction to Copulas

Author : Roger B. Nelsen
Publisher : Springer Science & Business Media
ISBN 13 : 1475730764
Total Pages : 227 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis An Introduction to Copulas by : Roger B. Nelsen

Download or read book An Introduction to Copulas written by Roger B. Nelsen and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt: Copulas are functions that join multivariate distribution functions to their one-dimensional margins. The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. With nearly a hundred examples and over 150 exercises, this book is suitable as a text or for self-study. The only prerequisite is an upper level undergraduate course in probability and mathematical statistics, although some familiarity with nonparametric statistics would be useful. Knowledge of measure-theoretic probability is not required. Roger B. Nelsen is Professor of Mathematics at Lewis & Clark College in Portland, Oregon. He is also the author of "Proofs Without Words: Exercises in Visual Thinking," published by the Mathematical Association of America.