Shape-preserving estimators of copulas

• Grant number: 24/23204-5
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2025
• End date: March 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Statistics
• Principal Investigator: Pedro Alberto Morettin
• Grantee: João Victor Bastos de Freitas
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/02538-0 - Time series, wavelets, high dimensional data and applications, AP.TEM

Abstract

An important task is the establishment of dependency relationships between two or more variables, and copulas are valuable tools in this regard. In many situations, the marginal distributions are known (or can be estimated), while the joint distribution is unknown, or may be difficult to estimate. Simply put, a copula is a function (actually, a special distribution function) that connects marginal distributions to their joint distribution. Copulas have been applied in economics, finance, and others. In financial econometrics, a key challenge is modeling asset returns for the purpose of calculating risk measures, such as the Value-at-Risk (VaR). A common assumption is that the returns are Gaussian (as in the RiskMetrics methodology), but it is known that these returns exhibit heavy tails and high kurtosis. Furthermore, in order to calculate risk measures, it is necessary to obtain the covariance matrix of a portfolio containing a large number of assets, making the modeling of a vector of assets a complex task. These challenges have contributed to the increasing use of copulas in financial modeling. Various approaches have been employed in estimating copulas: parametric methods (maximum likelihood estimation and method of moments), non-parametric methods (empirical copulas and kernel-based estimation) and semi-parametric methods. Most studies aim to estimate copulas from independent samples of a random vector. Fermanian and Scaillet (2003) adress the case of time series and propose kernel-based estimators for stationary time series. For estimators using wavelets and independent and identically distributed data, see Genest et al. (2009), Autin et al. (2010) and Gayraud and Tribouley (2010). Copula estimators for time series using wavelets were discussed by Morettin et al. (2011). An alternative approach is presented in Morettin et al. (2010).In many situations, as in the works mentioned above, the resulting estimators are not copulas. This occurs because the estimators use density estimators and distribution functions that do not satisfy properties that these functions should satisfy. For instance, density estimators using kernels or wavelets can be negative, depending on the choice of kernel or wavelet basis. The aim of this project is to obtain non-parametric estimators for copulas based on wavelets, using the results of Cosma et al. (2007) to estimate copulas via Sklar Theorem, as done by Morettin et al. (2011), but now obtaining estimators that preserve the shape, i.e., satisfy the properties of a copula.