# Abstracts Research Seminar Winter Term 2013/14

## Marius Hofert: Statistical and computational aspects of nested Archimedean copulas and beyond

After an introduction to (Archimedean) copulas, the class of nested Archimedean copulas is presented. In particular, we focus on sampling algorithms and likelihood-based estimation, the corresponding computational problems, and other challenges. In a second part of the talk, we briefly present a new R package which aims at simplifying statistical simulation studies and which carefully deals with important tasks such as parallel computing, seeding, catching of warnings and errors, and measuring run time.

## Omiros Papaspiliopoulos: Optimal filtering and the dual process

(Authors: Omiros Papaspiliopoulos and Matteo Ruggiero)

We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special case of our framework is the Kalman filter and we can devise computable filters for models with Cox-Ingersoll-Ross and Wright-Fisher processes as signals. There are deep connections between our approach and probabilistic constructions in population genetics. However, the talk will focus on the underlying methodology and the numerics. The talk is based on an article that is to appear in Bernoulli, and on-going work.

## David Edwards: Some context-specific graphical models for discrete longitudinal data

A rich family of models for discrete longitudinal data, called acyclic probabilistic finite automata (APFA), was introduced by Ron et al (1998). An APFA may be represented as a directed multigraph, and embodies a set of context-specific conditional independence relations that may be read off the graph. Here we look at the methods from a statistical perspective. We show how likelihood ratio tests may be constructed using standard contingency table methods, and describe how the selection algorithm of Ron et al (1998) may be modified to minimize a penalized likelihood criterion such as AIC or BIC. This is joint work with Smitha Ankinakatte.

Reference: Ron, D., Y. Singer, and N. Tishby (1998). On the learnability and usage of acyclic finite automata. Journal of Computer and System Sciences 56, 133-152.

## Mike Smith: Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation

Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis–Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

## Maria Kalli: Bayesian Semiparametric vector autoregressive models

Vector autoregressive models (VARs) are the working horse for much macroeconomic forecasting. The two key reasons for their popularity are: their ability to describe the dynamic structure of many variables, and their ability to conditionally forecast potential future paths of specified subsets of those variables. Whether a classical or a Bayesian approach is adopted in VAR modelling most (if not all) models are linear with normally generated innovations. We propose a Bayesian semi-parametric approach which uses the Dirichlet process mixture to construct non-linear first order stationary multivariate VAR processes with non-Gaussian innovations. Our method will be applied to macroeconomic time series.

## Jim Griffin: Modelling Macroeconomic Time Series using Regression Models with Time-varying Sparsity

Regression models are often used for prediction in macroeconomic time series. This allows other economic variables to be included in the regression. Often these effects are time-varying and this has lead to the wide-spread use of time-varying parameter model. If many variables are included, some form of variable selection is needed. It is natural to assume that some variables may be important for prediction at some times but not others. In this talk, I will discuss a Bayesian approach that allows this time-varying variable selection by constructing a suitable prior. The approach will be illustrated by applications to inflation forecasting and equity premium prediction.

## Roberto Casarin: Bayesian Calibration and Combination of Predictive Distributions

(joint paper with Francesco Ravazzolo and Tilmann Gneiting)

We provide a Bayesian approach to predictive density combination and calibration which accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights. We build on the predictive density calibration and combination framework of Gneiting et al. (2007) and propose the use of mixtures of Beta densities for the calibration and combination. One advantage of the proposed nonparametric approach relies upon the flexibility of the infinite beta mixtures to achieve a flexible continuous deformation of the linear combination of predictive distributions. We discuss properties of the methodology in simulation exercises with fat tails and multi-modal processes, and apply to predict daily stock returns and daily maximum wind speed.

## Damir Filipovic: Linear-Rational Term Structure Models

We introduce the class of linear-rational term structure models, where the state price density is modeled such that bond prices become linear-rational functions of the current state. This class is highly tractable with several distinct advantages: i) ensures non-negative interest rates, ii) easily accommodates unspanned factors affecting volatility and risk premia, and iii) admits analytical solutions to swaptions. For comparison, affine term structure models can match either i) or ii), but not both simultaneously, and never iii). A parsimonious specification of the model with three term structure factors and one, or possibly two, unspanned factors has a very good fit to both interest rate swaps and swaptions since 1997. In particular, the model captures well the dynamics of the term structure and volatility during the recent period of near-zero interest rates.