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Iva Mihaylova (University of St. Gallen, Switzerland)

DOI: 10.4018/978-1-4666-5202-6.ch048

Top## Introduction

It has been more than fifty years since Abe Sklar introduced the concept “copula”, meaning “a connection”, in the context of probabilistic metric spaces. In the last decade, copulas have become a widely used statistical tool for dependence modeling applications. Genest, Gendron, & Bourdeau-Brien, (2009) reported 750,000 Google hits of the word “copula” in 2009, compared to 10,000 in 2003 (Mikosch, 2006).

The popularity of copulas is explained by the growing evidence (see, e.g., Jondeau, Poon, & Rockinger, 2007) that the dependence among the financial and economic phenomena is incompatible with the traditionally applied (multivariate) Gaussian distribution and coefficient of linear correlation. Finding an appropriate multivariate perspective of a concrete dependence modeling task without resorting to copulas may be challenging or even impossible, because the set of available parametric multivariate distributions is considerably smaller than the set of parametric univariate distributions. As commented by Joe (1997), the “study of multivariate distributions is not easy because one cannot just write down a family of functions and expect it to satisfy the necessary conditions for multivariate cumulative distribution functions [...].” (p. 3). Sklar’s (1959) theorem has defined the copula concept and laid the theoretical foundation for a separate, two-stage modeling of the stochastic variables’ marginal distributions and their joint dependence structure.

The usefulness of copulas consists in the fact that any of them can link any valid marginal distribution into a valid multivariate distribution. Consequently, it can be stated that copulas expand the set of possible (not automatically meaning “appropriate”) multivariate distributions and increase the probability of finding an optimal dependence model for the concretely analyzed data. Due to their flexibility, copulas have become a preferred statistical tool when the research goal is to explore dependence. As an influential earlier contribution can be mentioned Embrechts, McNeil, & Straumann (1999). A detailed overview of copula methods is provided in the books of Joe (1997) and Nelsen (2006), while the emphasis in McNeil, Frey, & Embrechts (2005) is on risk management. In the area of finance, Cherubini, Luciano, & Vecchiato (2004) concentrate predominantly on copula applications for derivatives pricing and credit risk analysis, complemented by Cherubini, Mulinacci, Gobbi, & Romagnoli (2011) with new convolution-based copulas.

Sklar (1959) defines the copula concept only for the static (unconditional) case. However, there is well-documented evidence that the dependence measures of observed phenomena in economics and finance have a time-varying behavior (see, e.g., Engle, 2002). Patton (2006) has proposed a conditional copula, an extension of Sklar’s theorem, which incorporates both marginal and joint conditional distributions, as well as a parametric model for the dynamics of the copula dependence parameter. This new copula model can describe the time-varying dependence of multivariate time series data, the main theme of the present chapter. Its central goal is to briefly introduce the most popular static copulas in finance and economics, as well as to concentrate on recent advances and trends, concerning time-varying copulas.

Semi-Parametric Copula Estimation: This concept means that the participating marginal distributions are estimated non-parametrically (for example, by the empirical distribution function), while the copula function is estimated parametrically. Initially, this approach was introduced by Chen and Fan (2006) . However, as noted by Patton (in press) , the authors’ useful theoretical results valid for static copulas do not hold for time-varying conditional copulas, and new inference methods should be developed. In addition, the opposite combination of a parametric estimation of the marginal distributions and a non-parametric estimation of the copula also belongs to the semi-parametric copula estimation category (see, e.g., Hafner & Reznikova, 2010 ).

Coefficient of Upper/Lower Tail Dependence: The probability that the marginal distribution of a random variable X 1 exceeds a high/low quantile threshold on condition that the marginal distribution of a random variable X 2 also exceeds the same high/low quantile threshold. Stated differently, this coefficient measures the strength of the dependence in the tails of the bivariate distribution F ( X 1 , X 2 ).

Rank Correlation: A scalar measure of dependence in the interval [-1, 1], the calculation of which is based on the ranks of the data observations alone.

Non-Parametric Copula Estimation: In the copula theory, this concept means that both the participating marginal distributions and the copula function are estimated non-parametrically.

Conditional Copula: The concept has been introduced by Patton (2006) as an extension of Sklar’s (1959) theorem and incorporates both marginal and joint conditional distributions.

Time Series Data: A sequence of chronologically ordered in time or space quantitative observations, for example asset prices, exchange rates, stock exchange indices, the dynamics of monetary aggregates, imports, or exports. There exist two types of time series data: univariate (only one variable is measured over time or space) and multivariate (involves simultaneous measurement of more than one variable).

Copula: A dependence model, defined as a multivariate distribution function over a hypercube with uniform marginal distributions. The usefulness of copulas consists in the fact that any of them can link any marginal distribution into a valid multivariate distribution. Having in mind that the set of available parametric multivariate distributions is considerably smaller than the set of the parametric univariate distributions, a conclusion can be drawn that copulas expand the set of possible multivariate distributions and consequently increase the probability of finding an optimal dependence model for the concretely analyzed data.

Coefficient of Linear Correlation: A scalar measure of linear dependence in the interval [-1, 1], also referred to as “Pearson correlation coefficient”. Technically, it is derived from the covariance figure divided by the product of the standard deviations.

Ranks: The ranks of data observations are obtained by sorting them in order and by subsequently replacing each of them by its relative position in the order.

Fully Parametric Estimation: In the copula theory, this concept means that both the participating marginal distributions and the copula function are estimated parametrically.

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