Browsing by Subject "Copula"

Bayesian methods offer a flexible and convenient probabilistic learning framework to extract interpretable knowledge from complex and structured data. Such methods can characterize dependencies among multiple levels of hidden variables and share statistical strength across heterogeneous sources. In the first part of this dissertation, we develop two dependent variational inference methods for full posterior approximation in non-conjugate Bayesian models through hierarchical mixture- and copula-based variational proposals, respectively. The proposed methods move beyond the widely used factorized approximation to the posterior and provide generic applicability to a broad class of probabilistic models with minimal model-specific derivations. In the second part of this dissertation, we design probabilistic graphical models to accommodate multimodal data, describe dynamical behaviors and account for task heterogeneity. In particular, the sparse latent factor model is able to reveal common low-dimensional structures from high-dimensional data. We demonstrate the effectiveness of the proposed statistical learning methods on both synthetic and real-world data.

The dissertation consists of four chapters that concern topics on copulas for high dimensions. Chapter 1 proposes a new general model for high dimension joint distributions of asset returns that utilizes high frequency data and copulas. The dependence between returns is decomposed into linear and nonlinear components, which enables the use of high frequency data to accurately measure and forecast linear dependence, and the use of a new class of copulas designed to capture nonlinear dependence among the resulting linearly uncorrelated residuals. Estimation of the new class of copulas is conducted using a composite likelihood, making the model feasible even for hundreds of variables. A realistic simulation study verifies that multistage estimation with composite likelihood results in small loss in efficiency and large gain in computation speed.

Chapter 2, which is co-authored with Professor Andrew Patton, presents new models for the dependence structure, or copula, of economic variables based on a factor structure. The proposed models are particularly attractive for high dimensional applications, involving fifty or more variables. This class of models generally lacks a closed-form density, but analytical results for the implied tail dependence can be obtained using extreme value theory, and estimation via a simulation-based method using rank statistics is simple and fast. We study the finite-sample properties of the estimation method for applications involving up to 100 variables, and apply the model to daily returns on all 100 constituents of the S\&P 100 index. We find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms.

Chapter 3, which is co-authored with Professor Andrew Patton, considers the estimation of the parameters of a copula via a simulated method of moments type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed approach naturally also nests method of moments and generalized method of moments estimators. Drawing on results for simulation based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of over-identifying restrictions as a goodness-of-fit test. The results apply to both $iid$ and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study.

Chapter 4, which is co-authored with Professor Andrew Patton, 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 modelling high dimension covariance matrices and on recent work on models for general time-varying distributions. Our use of copula-based models enable 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.

Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. In this dissertation, we address this difficulty for analyzing spatial point-referenced data and hierarchical data. Several models are introduced by generalizing the joint quantile regression model of Yang and Tokdar (2017) and characterizing different dependency structures via a copula model on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out prediction. Multiple copula families are discussed for modeling response data with tail dependence and/or tail asymmetry. An effective model comparison criterion is provided for selecting between models with different combinations of sets of predictors, marginal base distributions and copula models.

Extensive simulation studies and real applications are presented to illustrate substantial gains of the proposed models in inference quality, prediction accuracy and uncertainty quantification over existing alternatives. Through case studies, we highlight that the proposed models admit great interpretability and are competent in offering insightful new discoveries of response-predictor relationship at non-central parts of the response distribution. The effectiveness of the proposed model comparison criteria is verified with both empirical and theoretical evidence.