Vine Copulas as a Way to Describe and Analyze Multi-Variate Dependence in Econometrics: Computational Motivation and Comparison with Bayesian Networks and Fuzzy Approaches

In the last decade, vine copulas emerged as a new efficient techniques for describing and analyzing multi-variate dependence in econometrics; see, e.g., [1–3, 7, 9–11, 13, 14, 21]. Our experience has shown, however, that while these techniques have been successfully applied to many practical problems of econometrics, there is still a lot of confusion and misunderstanding related to vine copulas. In this paper, we provide a motivation for this new technique from the computational viewpoint. We show that other techniques used to described dependence – Bayesian networks and fuzzy techniques – can be viewed as a particular case of vine copulas. 1 Copulas – A Useful Tool in Econometrics: Motivations and Descriptions Need for studying dependence in econometrics. Many researchers have observed that economics is more complex than physics. In physics, many parameters, many phenomena are independent. As a result, we can observe (and thoroughly study) simple systems which can be described by a small number of parameters. Based on these simple systems, we can separately determine the laws that describe mechanics, electrodynamics, thermodynamics, etc., and then combine these laws to describe more complex phenomena. In contrast, in economics, most phenomena are interrelated. Thus, to numerically describe economic phenomena, we need to take into account several dependent parameters. So, in econometrics, studying dependence is of utmost importance. Statistical character of economic phenomena. An additional complexity of economics – as compared to physics – is that while most physical processes are 2 S. Sriboonchitta, J. Liu, V. Kreinovich, and H. T. Nguyen deterministic, in economics, we can only make statistical predictions. If we repeatedly drop the same object from the Leaning Tower of Pisa (as Galileo did), we will largely observe the exact same behavior every time. In contrast, if several very similar restaurants open in the same area, some of them will survive and some will not, and it is practically impossible to predict which will survive – at best, we can predict the probability of survival. We can deterministically predict the future trajectory of a spaceship, but we can, at best, make statistical predictions about the future values of a stock index. Conclusion: we need to study dependence between random variables. Because of the statistical character of economic phenomena, each parameter describing the economics is a random variables. Thus, the need to study dependence means that we need to study dependence between random variables. Simplest case when random variables are independent: reminder. In order to analyze how to describe dependence of random variables, let us recall how independent random variables can be described. In general, a random variable Xi can be described by its cumulative distribution function Fi(xi) def = Prob(Xi ≤ xi). If two random variables X1 and X2 are independent, this means that their joint distribution function F (x1, x2) def = Prob(X1 ≤ x1 &X2 ≤ x2) is equal to the product of the marginal distributions F1(x1) and F2(x2): F (x1, x2) = F1(x1) · F2(x2). Towards describing dependence between two random variables: the notion of a copula. In the independent case, general, the joint distribution function F (x1, x2) of two random variables X1 and X2 is equal to the product F1(x1) · F2(x2) of the marginal distributions. In general, when the random variables X1 and X2 are dependent, the joint distribution function F (x1, x2) is different from the product F1(x1)·F2(x2). It is reasonable to describe this general joint distribution in such a way that we will clearly see how different is the joint distribution from the independent case. In the independent case, F (x,x2) is the product of the marginal distributions F1(x1) and F2(x2); to describe deviations from this product, it make sense to consider more general combination functions, i.e., to consider expressions of the type F (x1, x2) = C(F1(x1), F2(x2)). (1) Such combination functions C(a, b) are known as copulas; see, e.g., [19, 26] (see also [1–3, 7, 9–11, 13, 14, 21]). The independence case corresponds to the product combination function C(a, b) = a · b. The more the combination function C(a, b) is different from the product, the more dependent are the random variables X1 and X2. Probability density function in terms of the copula. The expression for the probability density function f(x1, x2) = ∂F (x1, x2) ∂x1∂x2 in terms of the copula can be