Financial time-series models generally assume a parametric form for both the trends and the innovations. However, parametric models for the innovations often lack expressive capacity or flexibility to describe the features of the empirical data and, in particular, the distribution of extreme events. This observation motivates the use of a semi-parametric approach, in which the distribution of the innovations is directly learned from the training data. However, because the actual innovations are heavy-tailed, some specific non-parametric method is needed to guarantee that the tails of the return distribution are correctly modeled. Our approach for modeling the density of the innovations is based on the kernel density estimator framework (Silverman,1986). To improve the quality of the density approximation in the tails, we followWand et al.(1991) and perform the estimation of the density in a transformed space, where the transformed data are approximately Gaussian. The transformation function is based on a fit of a stable distribution
(Nolan, 2002) to the original data. Experiments on simulated data show the superiority of the back-transformed kernel estimator over standard and adaptive kernel methods (Silverman,1986) when the distribution of the data is heavy-tailed. Additionally, an iterative algorithm (SPE) is introduced for the estimation of semi-parametric financial time-series models in which the unknown innovation distribution is approximated using a back-transformed kernel method. SPE generates estimates of the model parameters which are very close to the ones obtained by the maximum likelihood method when the actual functional form of the innovations is known. A series of experiments with empirical data show that SPE provides very accurate estimates of the conditional return density, especially in the tails of the distribution.
In this chapter, we have considered unidimensional time-series models for the description of price changes of single financial assets. However, we may be interested in the construction of multivariate models which are able to describe the joint evolution of the prices of several financial stocks. In particular, we would like to extend the proposed semi-parametric method to higher-dimensions. This can be accomplished by using copula functions (Joe,1997). Copulas allow to link separate univariate models into a joint multivariate model. For this process to be successful, we need accurate copula methods that are able to learn the dependencies present in the data without suffering from significant overfitting problems. The following chapter presents a novel semi-parametric bivariate copula method that can be used for this task.
Chapter
3
Semi-parametric Bivariate
Archimedean Copulas
The theory of copulas provides a general framework for the construction of multivariate models with a specific dependence structure and specific univariate marginal distributions. Parametric copulas often lack expressive capacity to capture the complex dependencies that often appear in empirical data. By contrast, non-parametric copulas can have poor generalization performance because of overfitting. As an intermediate approach, we introduce a flexible semi-parametric bivariate Archimedean copula model that provides accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood function and a term that penalizes non-smooth solutions. The performance of the semi-parametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semi-parametric estimators of Archimedean copulas previously introduced in the literature, two flexible methods that can describe more general and non-Archimedean dependence structures and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semi-parametric approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.
3.1
Introduction
Many standard univariate models do not have a simple extension to two or higher dimensions. In practice, only a reduced number of parametric distributional families with a closed analytical form are available for modeling multivariate data. Some examples can be found in the family of elliptical distributions (Fang et al., 1990). This family includes the multivariate Gaussian, the multivariate Student’s t and the elliptically contoured stable distributions (Nolan, 2002). However, the elliptical family has a limited range of distributional shapes and often cannot provide an accurate fit for empirical multivariate data. One of the main limitations of elliptical
distributions is that they cannot model asymmetries in the data. Nevertheless, it is possible to design extensions of the elliptical family that incorporates skewness (Genton,2004).
The theory of copulas (Joe,1997) provides a framework for the construction of multivariate models by expressing the distribution of the data in a canonical form that models the marginals separately from the dependence structure of the data. Let (X1, . . . , Xd)Tbe a continuous random
vector that follows distribution F and let F have continuous marginal distributions F1, . . . , Fd,
where Fi is the marginal of Xi. Here, F and F1, . . . , Fd are cumulative distribution functions. A
theorem due toSklar(1959) states that there is a unique function C, denoted the copula of F, such that
F(x1, . . . , xd) = C [F1(x1), . . . , Fd(xd)] . (3.1)
Therefore, the joint distribution F is uniquely determined by its marginals F1, . . . , Fd and its
copula C, where C is a cumulative distribution function with uniform marginals defined in the d-dimensional unit hypercube. Multivariate probabilistic models can be built by first fitting a different univariate model for each marginal and then, learning a copula function that links the univariate specifications in a joint multivariate model. The first step is straightforward and can be implemented using standard methods for modeling univariate data. The second step requires the specification of copula models that are both flexible, so that they are able to capture complex dependence structures in the data, and robust, so that overfitting is avoided. Parametric copula models such as the Gaussian or the Student’s t copulas are robust, but they generally lack expressive capacity to represent the complex multivariate dependencies that can be found in real- world data. Non-parametric copula models can approximate arbitrarily complex dependencies when sufficient amounts of data are available (Fermanian and Scaillet, 2003). However, this high flexibility and the absence of any distributional assumption on the underlying copula makes non-parametric methods more prone to overfitting. In this chapter, we adopt a semi-parametric approach based on the family of bivariate Archimedean copulas (Nelsen, 2006) that aims to strike a balance between flexibility and robustness (Hern´andez-Lobato and Su´arez,2009).
Bivariate Archimedean copulas are a specific class of copulas that are uniquely determined by a unidimensional generator function. Parametric Archimedean copulas assume a particular functional form for this generator, which depends only on a few parameters (Nelsen, 2006). More flexible models can be obtained when the Archimedean generator is expressed in terms of a one-dimensional functional parameter as proposed byVandenhende and Lambert(2005),
Lambert (2007), Gagliardini and Gourieroux(2007) and Dimitrova et al. (2008). Following the latter approach, we express the Archimedean generator in terms of a latent function that is simpler to model than the generator itself. The latent function is then represented using a basis of natural cubic splines. The coefficients of the expansion in the spline basis are computed by maximizing an objective function that includes the log-likelihood of the model and a term that penalizes the curvature of the functional parameter. This form of regularization is particularly convenient because latent functions with low curvature generate smooth copulas, which are less prone to overfitting.
Experiments with simulated data, and data from different domains of application (financial asset returns (Yahoo! Finance,2008) and precipitation data (Razuvaev et al.,2008)) are carried out to assess the performance of the proposed semi-parametric estimator. In these experiments, the novel approach is compared with alternative methods for bivariate copula estimation. These
Chapter3. Semi-parametric Bivariate Archimedean Copulas 31
include estimators of Archimedean copulas based on (a) Bayesian B-splines (Lambert, 2007) and (b) GeD splines (Dimitrova et al., 2008); (c) a flexible copula model based on Bayesian mixtures of Gaussians (Attias,1999;Bishop,2006); (d) a non-parametric copula model based on Gaussian kernel density estimators (Duong and Hazelton,2003;Fermanian and Scaillet,2003); (e) parametric Gaussian and Student’s t copulas (Malevergne and Sornette,2006); (f) parametric skewed Student’s t copulas (Demarta and McNeil, 2005) and finally, (g) standard parametric Archimedean copulas (Nelsen,2006). The excellent overall performance of the proposed semi- parametric copula method in the problems investigated confirms the capacity of this approach to capture complex dependencies in the data while avoiding overfitting.
The rest of the chapter is organized as follows. Section3.2 introduces a parameterization of bivariate Archimedean copulas in terms of a novel latent function. Section3.3describes the method proposed for the estimation of this function given a bivariate copula sample. Section3.4
presents a complete experimental evaluation of the new semi-parametric copula and subsections
3.4.1,3.4.2and3.4.3describe the results obtained for simulated, financial and precipitation data, respectively. Finally, Section3.5contains a summary and a discussion.