Heavy Tails And Copulas: Topics In Dependence Modelling In Economics And Finance

Topics in Dependence Modelling in Economics and Finance

Rustam Ibragimov, Artem Prokhorov;;;

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789814689816
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

-->

This book offers a unified approach to the study of crises, large fluctuations, dependence and contagion effects in economics and finance. It covers important topics in statistical modeling and estimation, which combine the notions of copulas and heavy tails — two particularly valuable tools of today's research in economics, finance, econometrics and other fields — in order to provide a new way of thinking about such vital problems as diversification of risk and propagation of crises through financial markets due to contagion phenomena, among others. The aim is to arm today's economists with a toolbox suited for analyzing multivariate data with many outliers and with arbitrary dependence patterns. The methods and topics discussed and used in the book include, in particular, majorization theory, heavy-tailed distributions and copula functions — all applied to study robustness of economic, financial and statistical models, and estimation methods to heavy tails and dependence.

-->

Chapter 1

Introduction and Overview

In this chapter, we set the stage by defining the subject matter of the book and describing the main tools used to study it. We also outline the structure of the book.

1.1. Crises, contagion and other features of modern economic and financial data

Modern economics, finance, risk management and insurance deal with data that is correlated, heterogeneous and/or heavy-tailed in some usually unknown fashion. When we say that data is heavytailed, or fat-tailed, we mean that it has a large proportion of relatively big fluctuations, where ‘large’ and ‘relatively big’ refer to proportions and fluctuations that would characterize a normally distributed random variable. These large fluctuations tend to happen simultaneously across various markets, even though individual markets usually behave differently, i.e., are heterogeneous.

Consider, for example, stock market returns during October 2008. In only a few days between October 6 and 10, the S&P500 — a US stock price index of the 500 largest companies — lost about 15%. If S&P500 was normally distributed, this event would happen no more often than once in a million years. Now look at the other world markets. During the same week, the FTSE100 — a key European stock index — lost about 14%, while the Nikkei 225 — a key Asian stock index — lost about 21%. Similar, and even larger, drops happened earlier, for example on October 19, 1987, the so-called Black Monday, but within a single day (see, e.g., Stock and Watson (2006), Section 2.4). Using estimates of the mean and standard deviation of the indices, it is possible to show that if the returns were normally distributed, the probability of such drops would be of order 10−¹⁰⁷, i.e., no more than the inverse of a googol (10¹⁰⁰). One can also add that 10−¹⁰⁷ is much smaller than the probability of choosing a particular atom from all atoms in the observable universe as their number is estimated to be 10⁸⁰!

Similar to stock market returns, crucial deviations from normality are observed for many other key financial and economic indicators and variables, including income and wealth, losses from natural disasters, firm and city sizes, operational risks and many others (see, e.g., the reviews by, Embrechts et al. (1997); McNeil et al. (2005); Gabaix (2009); Ibragimov et al. (2015)). When the number of extreme events is abnormally high, we refer to such distributions as heavy-tailed and when such events coincide across seemingly independent markets and produce market crashes, we call this asymmetric tail dependence. Distributions of financial returns are typically asymmetric because the number of extreme negative events — abnormal drops — tends to be higher than the number of positive events — abnormal jumps.

Evidence of heavy-tailedness and asymmetric tail dependence have been amply documented in equity markets (see, e.g., Ang and Chen (2002); Longin and Solnik (2001)), in foreign exchange markets (see, e.g., Patton (2006); Ibragimov et al. (2013)), especially surrounding various crises such as the Latin American debt crisis of 1982, the Asian currency crisis of 1997, the North American subprime lending crisis of 2008, etc. (see, e.g., Rodriguez (2007); Horta et al. (2010)).

Tail dependence in financial markets often takes the form of financial contagion, which is usually described as periods when declining prices and increased volatility spread among economic and financial markets causing markets that usually have little or no correlation to behave very similarly, often contrary to the fundamentals (see, e.g., Hamao et al. (1990); Lin et al. (1994); Longin and Solnik (2001); Mierau and Mink (2013)). Incidents of unfounded contagion are puzzling because they imply some sort of irrationality on the part of market participants — they cannot be explained using standard risk management strategies and optimal portfolio choices. For this reason, traditional explanations were based on various types of market imperfections, such as liquidity and coordination problems, information asymmetry, information cost and performance compensation factors (see, e.g., Dungey et al. (2005); Dornbusch et al. (2000); Kyle and Xiong (2001), for surveys).

This book provides an econometric treatment of such events. That is, it seeks to build a general framework for analyzing such events using statistical methods and models of relevance to economics and finance. There will be no economic models of crises or contagion; instead, we will look at the distributional and dependence characteristics of financial and economic data that may give rise to the described behavior and at modern methods of statistical and econometric analysis suitable for such data. The aim is to provide a framework for thinking about contagion statistically and econometrically and to survey the state-of-the-art econometric tools used in the setting of tail-dependent, heterogenous, and heavy-tailed data.

The two key distributional features here are heavy-tails — to accommodate excessive volatility or excess kurtosis—and copulas—to model tail dependence and contagion. In other words, we will examine models and methods used for the analysis of multivariate economic and financial data, whose copulas accommodate non-zero tail dependence and whose univariate distributions are diverse, heavy tailed and have relatively small and possibly unequal values of the tail index.

1.2. Econometric tools for modern financial and economic data

1.2.1. Multivariate distributions and copulas

From an econometric point of view, the complicated nature of financial time series originates from the statistical properties of disturbances affecting financial markets. These properties are captured by their cumulative distribution functions, or cdf’s. Individual behavior of a single financial indicator is represented by a univariate cdf, while joint behavior of multiple indices — a particular focus of our analysis — is characterized by a multivariate cdf.

Let F_k : R → [0, 1], k = 1, . . . , d, be one-dimensional cdf’s, also known as marginal cdf’s or simply marginals, and let ξ₁, . . . , ξ_d be independent r.v.’s on some probability space (Ω,

, P) with

(ξ_k ≤ x_k) = F_k(x_k), x_k ∈ R, k = 1, . . . , d. A multivariate cdf F(x₁, . . . , x_d), x_i ∈ R, i = 1, . . . , d, with given marginals F_k, is a function satisfying the following conditions:

(a) F(x₁, . . . , x_d) =

(X₁ ≤ x₁, . . . , X_d ≤ x_d) for some r.v.’s X₁, . . . , X_d on a probability space (Ω,

, P);

(b) the one-dimensional marginal cdf’s ...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Preface
• Foreword
• Acknowledgments
• Contents
• 1. Introduction and Overview
• 2. Portfolio Diversification under Independent Fat Tailed Risks
• 3. From Independence to Dependence via Copulas and U-statistics
• 4. Limits of Diversification under Fat Tails and Dependence
• 5. Robustness of Econometric Methods to Copula Misspecification and Heavy Tails
• 6. Copula Tests Using Information Matrix
• 7. Summary and Conclusion
• Bibliography
• Index