PLAN DE CAPACITACIÓN

The analysis of the dependence structure between two or more variables is of impor- tance in a wide number of problems. Whereas dependence can be captured by the so-called dependence measures (Rényi, 1959; Schweizer et al., 1981) (e.g. Pearson’s correlation coefficient, Kendall’s tau or Spearman’s rho), an alternative approach is given by copulas (M. Sklar, 1959; A. Sklar, 1973).

Consider a random vectorX= (X₁, ...,X_d),with marginals CDFs ofX_i,i=1, ...,d beingF_i(X_i) =P r[X_i≤x_i]. The random vectorU= (U₁, ...,U_d)obtained by apply- ing the probability integral transform to each component8_{is distributed in[0, 1]}d

with uniform marginals. Its joint cumulative distributionC is the copula ofX:

C(u₁, ...,u_d) =P r[U₁≤u₁, ...,U_d≤u_d], that is,U∼C, the copula ofX.

The Sklar theorem (M. Sklar, 1959) provides the theoretical foundation for cop- ula applications. BeH a multivariate CDF of a random vectorX,H(x₁, ...,x_d) = P r[X₁≤x₁, ...,X_d ≤x_d]with marginsF_i(x_i) = [X_i≤x_i]. For everyx∈d_{,H can}

be expressed in terms of its margins and a copulaC:

H(x₁, ...,x_d) =C(F₁(x₁), ...,F_d(x_d)).

WhereasH entails all the univariate and multivariate information onX,C contains

all the information about the dependence between the componentsX₁, ...,X_d. Con-

versely, for a copulaC :[0, 1]d_{→[0, 1]andd marginsF}

i,C(F1(x1), ...,Fd(xd))is a

d−dimensional CDF with marginsF_i,i =1, ...,d. Importantly, ifF_i are continu- ous, on the Cartesian products of the ranges Ran(F₁)×...×Ran(F_d),C is uniquely defined. A general reference for a rigorous introduction, comprehensive of all the mathematical background and preliminaries to the Sklar theorem is given by (Nelsen,

8_I.e.(U

2007). Analogous results hold for densities (e.g. Nelsen, 2007), and for conditional distributions (e.g. A. Patton, 2013).

Since the analysis of dependence is the main purpose of copulas, their applicability involves a wide range of fields and analyses. From the influential paper of (Embrechts, McNeil et al., 2002), there have been extensive applications in finance, econometrics, risk management and actuarial sciences (see e.g. Bouyé et al., 2000; Cherubini, Luciano and Vecchiato, 2004; A. Patton, 2013). The main motivation for using copulas in finance comes from evidence of non-normality in the dependence (e.g. Malevergne et al., 2003, amongy many others) between asset returns (A. Patton, 2013). This is critical in several directions, especially in risk management. Application of copula methods in value-at-risk evaluations are among the earliest (e.g. Cherubini and Luciano, 2001; Embrechts, Höing et al., 2003). The use of copulas in the study of financial contagion became relevant in recent years: the complex connection and dependence between markets are strongly simplified by a normal-dependency assumption, and importantly the strength of dependencies increase during periods of crisis (Erb et al., 1994; A. J. Patton, 2006). Early studies in this direction are the switching copula model of (Rodriguez, 2007), focusing on the Asian and Mexican crises. This research direction closely relates to the recent explosions of credit derivatives products and multiple underlying products such as basket default swap and collateralized debt obligations. These pools of assets and liabilities are of high complexity whose dependence structure has been studied in a copula perspective in several papers (see e.g. D. X. Li, 1999; Jouanin et al., 2001; Laurent et al., 2005). More about derivative pricing using copulas can be found, in e.g. Cherubini, Luciano and Vecchiato, 2004. On the other hand, seminal applications in optimal portfolio decision are those of (e.g. A. J. Patton, 2004; Kole et al., 2007). Early contributions from (A. J. Patton, 2001b; A. J. Patton, 2001a; A. J. Patton, 2006) explore the use of copula in time-series modelling. Conditional on a set of past information, copula parameters are allowed for time-variation in an autoregressive way. The time-varying dynamics in parameters naturally represents a suitable feature for financial time-series data. In a similar perspective, Jondeau et al., 2006 provide a copula-based extension of the framework in (B. E. Hansen, 1994), first advocating a Copula-GARCH model class.

The the discussion (Sokolinskiy et al., 2011), hints the development of Publication IV. One-day-ahead forecasts extracted from the HAR model (Corsi, 2009) are compared with those of copula-based realized-volatility forecasts. This is achieved by decom-

posing the joint distribution of the integrated volatility estimator (RR) and its first

lag into marginal distributionsF (estimated non-parametrically) and a copula density

c(estimated via maximum likelihood). The conditioning the density f ofRR_t on

RR_t−1: fRR_t|RR_t−1=cF(RR_t),FRR_t−1f(RR_t), forecasts are obtained via Monte Carlo (MC) simulation. (i) Draws of today’s volatility (captured with the realized range estimator) are simulated from the fitted copula given yesterday’s volatil- ity (its lagged value) (ii) the uniform draws are transformed via inverse empirical distribution function (iii) their mean is taken as a conditional forecast of today’s volatility given yesterday’s. Under a Gumbel copula, their specification is shown to beat the HAR model in out-of-sample analyses. The authors also implement a conditional-copula version of their models (relying on A. J. Patton, 2006) with time- varying parameters which however seems not to improve the accuracy of volatility forecasts. A generalization of their model to account for a four-dimensional copula that resembles the modeling spirit of the HAR model (dealing with four volatility variables), and that possibly allows for an evaluation of the conditional expectation without relying on MC methods is the task developed in Publication IV. However, the multidimensional extension of bivariate copula models is not immediate. For instance, long-memory properties observed in volatility would suggest that the de- pendence strength and type (as empirically observed) are different among pairs of variables involving volatility measures at different scales: a multivariate copula that is flexible in this concern would represent an attractive solution.

Extending of copula-based models in high dimension is most obvious but difficult problem (A. Patton, 2013). Flexible yet parsimonious and feasible directions are Factor-copulas (D. H. Oh et al., 2017) and Vine-copulas (see Section 4.3.2). For a comprehensive overview on copula construction methods see (e.g. Joe, 2014, chapter 3). Vine copulas applications are widespread, examples are risk-management and value-at-risk applications (e.g. Weiß et al., 2013; Reboredo et al., 2015), volatility modelling (e.g. Vaz de Melo Mendes et al., 2014; So et al., 2014; E. C. Brechmann, Heiden et al., 2018) - but under a HAR-like approach, and in the analysis of financial returns (e.g. Chollete et al., 2009; Nikoloulopoulos et al., 2012; Dissmann et al., 2013; Joe, 2014).

3

DATA

The two datasets utilized in the publications are here introduced to the reader. Refer to the Publications I-IV for further details on variables and time-series that each of them deals with.