This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.
In this paper we develop a Bayesian procedure for estimating multivariate stochastic volatility (MSV) using state space models. A multiplicative model based on inverted Wishart and multivariate singular beta distributions is proposed for the evolution of the volatility, and a flexible sequential volatility updating is employed. Being computationally fast, the resulting estimation procedure is particularly suitable for on-line forecasting. Three performance measures are discussed in the context of model selection: the log-likelihood criterion, the mean of standardized one-step forecast errors, …
Multivariate stochastic volatility models with skew distributions are proposed. Exploiting Cholesky stochastic volatility modeling, univariate stochastic volatility processes with leverage effect and generalized hyperbolic skew t-distributions are embedded to multivariate analysis with time-varying correlations. Bayesian prior works allow this approach to provide parsimonious skew structure and to easily scale up for high-dimensional problem. Analyses of daily stock returns are illustrated. Empirical results show that the time-varying correlations and the sparse skew structure contribute to im…
The stochastic volatility model is a popular tool for modeling the volatility of assets. The model is a nonlinear and non-Gaussian state space model, and consequently is difficult to fit. Many approaches, both classical and Bayesian, have been developed that rely on numerically intensive techniques such as quasi-maximum likelihood estimation and Markov chain Monte Carlo (MCMC). Convergence and mixing problems still plague MCMC algorithms when drawing samples sequentially from the posterior distributions. While particle Gibbs methods have been successful when applied to nonlinear or non-Gaussia…
We formulate a discrete-time Bayesian stochastic volatility model for high-frequency stock-market data that directly accounts for microstructure noise, and outline a Markov chain Monte Carlo algorithm for parameter estimation. The methods described in this paper are designed to be coherent across all sampling timescales, with the goal of estimating the latent log-volatility signal from data collected at arbitrarily short sampling periods. In keeping with this goal, we carefully develop a method for eliciting priors. The empirical results derived from both simulated and real data show that dire…
This paper discusses the efficient Bayesian estimation of a multivariate factor stochastic volatility (Factor MSV) model with leverage. We propose a novel approach to construct the sampling schemes that converges to the posterior distribution of the latent volatilities and the parameters of interest of the Factor MSV model based on recent advances in Particle Markov chain Monte Carlo (PMCMC). As opposed to the approach of Chib et al. (2006} and Omori et al. (2007}, our approach does not require approximating the joint distribution of outcome and volatility innovations by a mixture of bivariate…
We discuss efficient Bayesian estimation of dynamic covariance matrices in multivariate time series through a factor stochastic volatility model. In particular, we propose two interweaving strategies (Yu and Meng, Journal of Computational and Graphical Statistics, 20(3), 531-570, 2011) to substantially accelerate convergence and mixing of standard MCMC approaches. Similar to marginal data augmentation techniques, the proposed acceleration procedures exploit non-identifiability issues which frequently arise in factor models. Our new interweaving strategies are easy to implement and come at almo…
The recently introduced class of simultaneous graphical dynamic linear models (SGDLMs) defines an ability to scale on-line Bayesian analysis and forecasting to higher-dimensional time series. This paper advances the methodology of SGDLMs, developing and embedding a novel, adaptive method of simultaneous predictor selection in forward filtering for on-line learning and forecasting. The advances include developments in Bayesian computation for scalability, and a case study in exploring the resulting potential for improved short-term forecasting of large-scale volatility matrices. A case study co…
Jump stochastic volatility models are central to financial econometrics for volatility forecasting, portfolio risk management, and derivatives pricing. Markov Chain Monte Carlo (MCMC) algorithms are computationally unfeasible for the sequential learning of volatility state variables and parameters, whereby the investor must update all posterior and predictive densities as new information arrives. We develop a particle filtering and learning algorithm to sample posterior distribution in Merton's jump stochastic volatility. This allows to filter spot volatilities and jump times, together with se…
Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in deta…
Financial markets for Liquified Natural Gas (LNG) are an important and rapidly-growing segment of commodities markets. Like other commodities markets, there is an inherent spatial structure to LNG markets, with different price dynamics for different points of delivery hubs. Certain hubs support highly liquid markets, allowing efficient and robust price discovery, while others are highly illiquid, limiting the effectiveness of standard risk management techniques. We propose a joint modeling strategy, which uses high-frequency information from thickly-traded hubs to improve volatility estimation…
This paper estimates models of high frequency index futures returns using `around the clock' 5-minute returns that incorporate the following key features: multiple persistent stochastic volatility factors, jumps in prices and volatilities, seasonal components capturing time of the day patterns, correlations between return and volatility shocks, and announcement effects. We develop an integrated MCMC approach to estimate interday and intraday parameters and states using high-frequency data without resorting to various aggregation measures like realized volatility. We provide a case study using …
The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work, however, has developed methods that couple model and parameter moves, first through reversible jump methods a…
The sampling efficiency of MCMC methods in Bayesian inference for stochastic volatility (SV) models is known to highly depend on the actual parameter values, and the effectiveness of samplers based on different parameterizations varies significantly. We derive novel algorithms for the centered and the non-centered parameterizations of the practically highly relevant SV model with leverage, where the return process and innovations of the volatility process are allowed to correlate. Moreover, based on the idea of ancillarity-sufficiency interweaving (ASIS), we combine the resulting samplers in o…
Financial studies require volatility based models which provides useful insights on risks related to investments. Stochastic volatility models are one of the most popular approaches to model volatility in such studies. The asset returns under study may come in multiple clusters which are not captured well assuming standard distributions. Mixture distributions are more appropriate in such situations. In this work, an algorithm is demonstrated which is capable of studying finite mixtures but with unknown number of components. This algorithm uses a Birth-Death process to adjust the number of comp…
Local volatility is an important quantity in option pricing, portfolio hedging, and risk management. It is not directly observable from the market; hence calibrations of local volatility models are necessary using observable market data. Unlike most existing point-estimate methods, we cast the large-scale nonlinear inverse problem into the Bayesian framework, yielding a posterior distribution of the local volatility, which naturally quantifies its uncertainty. This extra uncertainty information enables traders and risk managers to make better decisions. To alleviate the computational cost, we …
We propose a two stage procedure for the estimation of the parameters of a fairly general, continuous-time stochastic volatility. An important ingredient of the proposed method is the Cuchiero-Teichmann volatility estimator, which is based on Fourier transforms and provides a continuous time estimate of the latent process. This estimate is then used to construct an approximate likelihood for the parameters of interest, whose restrictions are taken into account through prior distributions. The procedure is shown to be highly successful for constructing the posterior distribution of the paramete…
Vector autoregressions (VARs) with multivariate stochastic volatility are widely used for structural analysis. Often the structural model identified through economically meaningful restrictions--e.g., sign restrictions--is supposed to be independent of how the dependent variables are ordered. But since the reduced-form model is not order invariant, results from the structural analysis depend on the order of the variables. We consider a VAR based on the factor stochastic volatility that is constructed to be order invariant. We show that the presence of multivariate stochastic volatility allows …
The leverage effect-- the correlation between an asset's return and its volatility-- has played a key role in forecasting and understanding volatility and risk. While it is a long standing consensus that leverage effects exist and improve forecasts, empirical evidence paradoxically do not show that most individual stocks exhibit this phenomena, mischaracterizing risk and therefore leading to poor predictive performance. We examine this paradox, with the goal to improve density forecasts, by relaxing the assumption of linearity in the leverage effect. Nonlinear generalizations of the leverage e…
Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a Cholesky decomposition for the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this Cholesky decomposition. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We deve…
In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution $F_{true}$. First, the Dirichlet process is constructed on the unknown marginal distributions of $F_{true}$. Then a Gaussian copula model is utilized to capture the dependence structure of $F_{true}$. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Several interesting theoretical results of the approach are derived. Finally, through several simulated examples and a real data set, the proposed approach…
As the dynamic structure of the financial markets is subject to dramatic changes, a model capable of providing consistently accurate volatility estimates must not make strong assumptions on how prices change over time. Most volatility models impose a particular parametric functional form that relates an observed price change to a volatility forecast (news impact function). We propose a new class of functional coefficient semiparametric volatility models where the news impact function is allowed to be any smooth function, and study its ability to estimate volatilities compared to the well known…
We take a new look at the problem of disentangling the volatility and jumps processes of daily stock returns. We first provide a computational framework for the univariate stochastic volatility model with Poisson-driven jumps that offers a competitive inference alternative to the existing tools. This methodology is then extended to a large set of stocks for which we assume that their unobserved jump intensities co-evolve in time through a dynamic factor model. To evaluate the proposed modelling approach we conduct out-of-sample forecasts and we compare the posterior predictive distributions ob…
Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A Lévy process with infinite jump activity and microstructure noise is considered one of the simplest, yet accurate enough, models for financial data at high-frequency. Utilizing this model, we propose a "purposely misspecified" posterior of the volatility obtained by ignoring the jump-component of the process. The misspecified posterior is further corrected by a simple estimate of the location shift and re-scaling of the log likelihood. Our main result establishes a Bernst…
We present a Bayesian approach for modeling multivariate, dependent functional data. To account for the three dominant structural features in the data--functional, time dependent, and multivariate components--we extend hierarchical dynamic linear models for multivariate time series to the functional data setting. We also develop Bayesian spline theory in a more general constrained optimization framework. The proposed methods identify a time-invariant functional basis for the functional observations, which is smooth and interpretable, and can be made common across multivariate observations for …
In this paper we take up Bayesian inference in general multivariate stable distributions. We exploit the representation of Matsui and Takemura (2009) for univariate projections, and the representation of the distributions in terms of their spectral measure. We present efficient MCMC schemes to perform the computations when the spectral measure is approximated discretely or, as we propose, by a normal distribution. Appropriate latent variables are introduced to implement MCMC. In relation to the discrete approximation, we propose efficient computational schemes based on the characteristic funct…
Several studies explore inferences based on stochastic volatility (SV) models, taking into account the stylized facts of return data. The common problem is that the latent parameters of many volatility models are high-dimensional and analytically intractable, which means inferences require approximations using, for example, the Markov Chain Monte Carlo or Laplace methods. Some SV models are expressed as a linear Gaussian state-space model that leads to a marginal likelihood, reducing the dimensionality of the problem. Others are not linearized, and the latent parameters are integrated out. How…
Change detection involves segmenting sequential data such that observations in the same segment share some desired properties. Multivariate change detection continues to be a challenging problem due to the variety of ways change points can be correlated across channels and the potentially poor signal-to-noise ratio on individual channels. In this paper, we are interested in locating additive outliers (AO) and level shifts (LS) in the unsupervised setting. We propose ABACUS, Automatic BAyesian Changepoints Under Sparsity, a Bayesian source separation technique to recover latent signals while al…
A Bayesian procedure is developed for multivariate stochastic volatility, using state space models. An autoregressive model for the log-returns is employed. We generalize the inverted Wishart distribution to allow for different correlation structure between the observation and state innovation vectors and we extend the convolution between the Wishart and the multivariate singular beta distribution. A multiplicative model based on the generalized inverted Wishart and multivariate singular beta distributions is proposed for the evolution of the volatility and a flexible sequential volatility upd…
Multivariate functional data arise in a wide range of applications. One fundamental task is to understand the causal relationships among these functional objects of interest, which has not yet been fully explored. In this article, we develop a novel Bayesian network model for multivariate functional data where the conditional independence and causal structure are both encoded by a directed acyclic graph. Specifically, we allow the functional objects to deviate from Gaussian process, which is adopted by most existing functional data analysis models. The more reasonable non-Gaussian assumption i…
Capturing the potentially strong dependence among the peak concentrations of multiple air pollutants across a spatial region is crucial for assessing the related public health risks. In order to investigate the multivariate spatial dependence properties of air pollution extremes, we introduce a new class of multivariate max-stable processes. Our proposed model admits a hierarchical tree-based formulation, in which the data are conditionally independent given some latent nested $α$-stable random factors. The hierarchical structure facilitates Bayesian inference and offers a convenient and inter…
We perform a Bayesian analysis of the p-variate skew-t model, providing a new parameterization, a set of non-informative priors and a sampler specifically designed to explore the posterior density of the model parameters. Extensions, such as the multivariate regression model with skewed errors and the stochastic frontiers model, are easily accommodated. A novelty introduced in the paper is given by the extension of the bivariate skew-normal model given in Liseo & Parisi (2013) to a more realistic p-variate skew-t model. We also introduce the R package mvst, which allows to estimate the multiva…
Bayesian methods and software for spatial data analysis are generally now well established in the scientific community. Despite the wide application of spatial models, the analysis of multivariate spatial data using R-INLA has not been widely described in the existing literature. Therefore, the main objective of this article is to demonstrate that R-INLA is a convenient toolbox to analyse different types of multivariate spatial datasets. Additionally, this will be illustrated by analysing three datasets which are publicly available. Furthermore, the details and the R code of these analyses are…
This paper proposes factor stochastic volatility models with skew error distributions. The generalized hyperbolic skew t-distribution is employed for common-factor processes and idiosyncratic shocks. Using a Bayesian sparsity modeling strategy for the skewness parameter provides a parsimonious skew structure for possibly high-dimensional stochastic volatility models. Analyses of daily stock returns are provided. Empirical results show that the skewness is important for common-factor processes but less for idiosyncratic shocks. The sparse skew structure improves prediction and portfolio perform…
Stochastic Volatility in Mean Models with Heavy Tails: A Fast Approximate Bayesian Inference Using Hidden Markov Models
University of São Paulo, Instituto de Ciências Matemáticas e de Computação（圣保罗大学，数学与计算科学研究所）；Federal University of Rio de Janeiro, Department of Statistics（里约热内卢联邦大学，统计学系）；Pontificia Universidad Católica del Perú, Department of Economics（秘鲁天主教 Pontificia 大学，经济学系）
Multivariate categorical data are common in many fields. We are motivated by election polls studies assessing evidence of changes in voters opinions with their candidates preferences in the 2016 United States Presidential primaries or caucuses. Similar goals arise routinely in several applications, but current literature lacks a general methodology which combines flexibility, efficiency, and tractability in testing for group differences in multivariate categorical data at different---potentially complex---scales. We address this goal by leveraging a Bayesian representation which factorizes the…
A Bayesian approach is developed to analyze change points in multivariate time series and space-time data. The methodology is used to assess the impact of extended inundation on the ecosystem of the Gulf Plains bioregion in northern Australia. The proposed approach can be implemented for dynamic mixture models that have a conditionally Gaussian state space representation. Details are given on how to efficiently implement the algorithm for a general class of multivariate time series and space-time models. This efficient implementation makes it feasible to analyze high dimensional, but of realis…
We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for defining distributions on sparsity structure of random eigenmatrices. We explore theoretical aspects and implications for conditional independence structures arising in multivariate Gaussian models, and discuss connections with sparse PCA, factor analysis and Gaussian graphical models. Methodology includes model-based exploratory data analysis and Bayesian analysis…
This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non-parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe…
Joint modeling of spatially-oriented dependent variables is commonplace in the environmental sciences, where scientists seek to estimate the relationships among a set of environmental outcomes accounting for dependence among these outcomes and the spatial dependence for each outcome. Such modeling is now sought for massive data sets with variables measured at a very large number of locations. Bayesian inference, while attractive for accommodating uncertainties through hierarchical structures, can become computationally onerous for modeling massive spatial data sets because of its reliance on i…
Factor analysis is a flexible technique for assessment of multivariate dependence and codependence. Besides being an exploratory tool used to reduce the dimensionality of multivariate data, it allows estimation of common factors that often have an interesting theoretical interpretation in real problems. However, standard factor analysis is only applicable when the variables are scaled, which is often inappropriate, for example, in data obtained from questionnaires in the field of psychology,where the variables are often categorical. In this framework, we propose a factor model for the analysis…
High resolution geospatial data are challenging because standard geostatistical models based on Gaussian processes are known to not scale to large data sizes. While progress has been made towards methods that can be computed more efficiently, considerably less attention has been devoted to big data methods that allow the description of complex relationships between several outcomes recorded at high resolutions by different sensors. Our Bayesian multivariate regression models based on spatial multivariate trees (SpamTrees) achieve scalability via conditional independence assumptions on latent r…
We consider the problem of multivariate density deconvolution where the distribution of a random vector needs to be estimated from replicates contaminated with conditionally heteroscedastic measurement errors. We propose a conceptually straightforward yet fundamentally novel and highly robust approach to multivariate density deconvolution by stochastically rotating the replicates toward the corresponding true latent values. We also address the additionally significantly challenging problem of accommodating conditionally heteroscedastic measurement errors in this newly introduced framework. We …
We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcom…
We present a Bayesian mixture model for estimating the joint distribution of mixed ordinal, nominal, and continuous data conditional on a set of fixed variables. The model uses multivariate normal and categorical mixture kernels for the random variables. It induces dependence between the random and fixed variables through the means of the multivariate normal mixture kernels and via a truncated local Dirichlet process. The latter encourages observations with similar values of the fixed variables to share mixture components. Using a simulation of data fusion, we illustrate that the model can est…
