**Organizing Committee**: José Blanchet, Mark Brown, Lorán Chollete, and Victor de la Peña

**DATE: ***June 7 – 10, 2016 *

**LOCATION: ***Columbia University, Statistics Department*

*(1255 Amsterdam Avenue, Room 903, 9 ^{th} Floor)*

**Dependence Workshop: June 7 – 8 **(11:00 AM – 1:00 PM and 2:30 PM – 4:30 PM)

Instructor, Professor **Alex McNeil** http://www.macs.hw.ac.uk/~mcneil/

**Thursday, June 9: Session 1**

10:30 – 11:30 **Alex McNeil** http://www.macs.hw.ac.uk/~mcneil/

11:30 – 12:30 ** Albert Marshall **http://www.stat.ubc.ca/People/Home/index.php?person=:marshall

Title: “Let the marginals select the copula”

Abstract: Several examples are discussed where properties or origins of marginal distributions are used to determine a joint distribution. This is work done jointly with Ingram Olkin, some old and some new. The focus of the new work is the Gompertz distribution, where bivariate versions are found using both functional and differential equations. Included are discussions of motivations that lead to the work.

12:45 – 1:45 Lunch

2:00 – 3:00 **Memorial Session for Ingram Olkin**

3:00 – 3:30 Coffee

3:30 – 4:30 ** Andrew Patton ** http://www.stern.nyu.edu/faculty/bio/andrew-patton

Title: “Time-Varying Systemic Risk: Evidence from a Dynamic Copula Model of CDS Spreads”

Abstract: This paper proposes a new class of copula-based dynamic models for high dimension conditional distributions, facilitating the estimation of a wide variety of measures of systemic risk. Our proposed models draw on successful ideas from the literature on modeling high dimension covariance matrices and on recent work on models for general time-varying distributions. Our use of copula-based models enables the estimation of the joint model in stages, greatly reducing the computational burden. We use the proposed new models to study a collection of daily credit default swap (CDS) spreads on 100 U.S. firms over the period 2006 to 2012. We find that while the probability of distress for individual firms has greatly reduced since the financial crisis of 2008-09, the joint probability of distress (a measure of systemic risk) is substantially higher now than in the pre-crisis period.

**Dinner**

**Friday, June 10: Session 2**

10:30 – 11:30 ** Harry Joe **http://www.stat.ubc.ca/~harry/

Title: “Factor copula model constructions”

Abstract: Classical Gaussian factor models can be extended to conditional independence models where observed variables (not necessarily Gaussian distributed) are conditionally independent given one or more latent variables.

There are several constructions of factor models in statistical finance that can allow for tail dependence relative to Gaussian. One general class of factor models is based on vines with latent variables; these can accommodate non-Gaussian common factor and structured factor models, including bi-factor and nested structures. Models based on vines have bivariate copulas as building blocks — some copulas apply to two univariate margins in the first tree of the vine, and other copulas apply to two univariate conditional distributions in higher order trees.

Another construction consists of Laplace-transform-Archimedean comonotonic factor models. This extends the Marshall-Olkin construction of exchangeable Laplace-transform-Archimedean copulas to more flexible factor models with one Laplace transform per observed variable that links to the unobserved (resilience) variable.

Some applications and numerical methods for the factor copula models will be mentioned.

11:30 – 12:30 * Gabor Szekely *http://www-math.bgsu.edu/~gabors/

Title: “Axioms of Dependence Measures: Distance Correlation and Decorrelation”

Abstract: Rényi (1959) proposed a set of seven axioms for dependence measures but even if a dependence measure satisfies all his very restrictive axioms, the dependence measure is not interpretable, not even the value 1 (in other words we do not have an if and only if characterization of the property: dependence measure = 1). In this talk we propose three very simple axioms and show that the distance correlation introduced by the speaker ten years ago satisfies all these axioms. Noticeably absent are axioms such as symmetry, invariance with respect to 1-1 transformations of the range space, equality to absolute correlation in the joint Gaussian setting. — In the second part of the talk (joint work with T. F. Mori) we show that every random vector X is equal in distribution to univariate functions f(t), g(t),… of “universal”, uncorrelated, real valued random variables. If X has zero expected value, finite variance and absolutely continuous distribution then these functions can be chosen to be +t if |t| is in a given Borel set and –t otherwise. For bivariate Gaussian X the functions f and g do not even depend on the possibly unknown correlation. Is this true for all d-variate Gaussian variables even if d > 2 ? — Finally, if X and Y are uncorrelated random variables under the conditions a < X < A and b< Y < B for all a < A and b < B real numbers then X and Y are independent.

12:45 – 1:45 Lunch

2:00 – 3:00 ** Bodhisattva Sen** http://www.stat.columbia.edu/~bodhi/Bodhi/Welcome.html

Title: “Two Applications of Testing for Mutual Independence”

Abstract: The talk will focus on two applications of testing for mutual independence: (a) in a goodness-of-fit test of a linear model, and (b) testing for conditional independence.

In the first part of the talk we consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and the predictor variables, and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the residual obtained from the parametric fit and the predictor using the Hilbert—Schmidt independence criterion (Gretton et al. (2008)). We develop distribution theory for the proposed test statistic, both under the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution.

In the second part of the talk, given a continuous random vector (X,Z), we define a notion of a nonparametric residual of X on Z that is always independent of the predictor Z. We study its properties and show that the proposed notion of residual matches with the usual residual (error) in a multivariate normal regression model. Given a random vector (X,Y,Z), we use this notion of residual to show that the conditional independence between X and Y, given Z, is equivalent to the mutual independence of the residuals (of X on Z and Y on Z) and Z. This result is used to develop a test for conditional independence. We propose a bootstrap scheme to approximate the critical value of this test.

**Reception**

**ATTENDEES and SPEAKERS**

- Jose Blanchet (Columbia University)
- Mark Brown (Columbia University)
- Lorán Chollete (University of St Andrews)
- Victor de la Peña (Columbia University)
- Harry Joe (University of British Columbia)
- Alex McNeil (Heriot Watt University)
- Albert Marshall (University of British Columbia)
- Andrew Patton (New York University)
- Bodhisattva Sen (Columbia University)
- Gabor Szekely (National Science Foundation)