Threshold variable selection of asymmetric stochastic volatility models

A threshold stochastic volatility (SV) model is used for capturing time-varying volatilities and nonlinearity. Two adaptive Markov chain Monte Carlo (MCMC) methods of model selection are designed for the selection of threshold variables for this family of SV models. The first method is the direct estimation which approximates the model posterior probabilities of competing models. Using parallel MCMC sampling to estimate these probabilities, the best threshold variable is selected with the highest posterior model probability. The second method is to use the deviance information criterion to compare among these competing models and select the best one. Simulation results lead us to conclude that for large samples the posterior model probability approximation method can give an accurate approximation of the posterior probability in Bayesian model selection. The method delivers a powerful and sharp model selection tool. An empirical study of five Asian stock markets provides strong support for the threshold variable which is formulated as a weighted average of important variables.     

随机波动模型的马尔可夫链—蒙特卡洛模拟方法——在沪市收益率序列上的应用

针对具有Markov区制转移的、波动均值状态相依的随机波动模型,基于贝叶斯分析,我们推导并给出了对区制转移随机波动模型的MCMC估计方法,其中对参数估计采用Gibbs抽样方法,对潜在对数波动和区制的状态变量估计采用"向前滤波、向后抽样"的多步移动方法;利用该模型,对我国上证综指周收益率进行了实证分析,发现对沪市波动性有较好的描述,捕捉了波动的时变性、聚类性和非线性特征,同时刻画了沪市的高低波动状态转换过程。     

MCMC calibration of spot‐prices models in electricity markets

The calibration of some stochastic differential equation used to model spot prices in electricity markets is investigated. As an alternative to relying on standard likelihood maximization, the adoption of a fully Bayesian paradigm is explored, that relies on Markov chain Monte Carlo (MCMC) stochastic simulation and provides the posterior distributions of the model parameters. The proposed method is applied to one‐ and two‐factor stochastic models, using both simulated and real data. The results demonstrate good agreement between the maximum likelihood and MCMC point estimates. The latter approach, however, provides a more complete characterization of the model uncertainty, an information that can be exploited to obtain a more realistic assessment of the forecasting error. In order to further validate the MCMC approach, the posterior distribution of the Italian electricity price volatility is explored for different maturities and compared with the corresponding maximum likelihood estimates.     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

Bayesian semiparametric Markov switching stochastic volatility model

This paper proposes a novel Bayesian semiparametric stochastic volatility model with Markov switching regimes for modeling the dynamics of the financial returns. The distribution of the error term of the returns is modeled as an infinite mixture of Normals; meanwhile, the intercept of the volatility equation is allowed to switch between two regimes. The proposed model is estimated using a novel sequential Monte Carlo method called particle learning that is especially well suited for state‐space models. The model is tested on simulated data and, using real financial times series, compared to a model without the Markov switching regimes. The results show that including a Markov switching specification provides higher predictive power for the entire distribution, as well as in the tails of the distribution. Finally, the estimate of the persistence parameter decreases significantly, a finding consistent with previous empirical studies.     

Quantile forecasting based on a bivariate hysteretic autoregressive model with GARCH errors and time ‐varying correlations

To understand and predict chronological dependence in the second‐order moments of asset returns, this paper considers a multivariate hysteretic autoregressive (HAR) model with generalized autoregressive conditional heteroskedasticity (GARCH) specification and time‐varying correlations, by providing a new method to describe a nonlinear dynamic structure of the target time series. The hysteresis variable governs the nonlinear dynamics of the proposed model in which the regime switch can be delayed if the hysteresis variable lies in a hysteresis zone. The proposed setup combines three useful model components for modeling economic and financial data: (1) the multivariate HAR model, (2) the multivariate hysteretic volatility models, and (3) a dynamic conditional correlation structure. This research further incorporates an adapted multivariate Student t innovation based on a scale mixture normal presentation in the HAR model to tolerate for dependence and different shaped innovation components. This study carries out bivariate volatilities, Value at Risk, and marginal expected shortfall based on a Bayesian sampling scheme through adaptive Markov chain Monte Carlo (MCMC) methods, thus allowing to statistically estimate all unknown model parameters and forecasts simultaneously. Lastly, the proposed methods herein employ both simulated and real examples that help to jointly measure for industry downside tail risk.     

Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity

Compared to the conditional mean or median, conditional quantiles provide a more comprehensive picture of a variable in various scenarios. A semi-parametric quantile estimation method for a double threshold auto-regression with exogenous regressors and heteroskedasticity is considered, allowing representation of both asymmetry and volatility clustering. As such, GARCH dynamics with nonlinearity are added to a nonlinear time series regression model. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between the quantile loss function and the asymmetric-Laplace distribution, is employed for estimation and inference, simultaneously estimating and accounting for nonlinear heteroskedasticity plus unknown threshold limits and delay lags. A simulation study illustrates sampling properties of the method. Two data sets are considered in the empirical applications: modelling daily maximum temperatures in Melbourne, Australia; and exploring dynamic linkages between financial markets in the US and Hong Kong.     

Model selection of a switching mechanism for financial time series

The threshold autoregressive model with generalized autoregressive conditionally heteroskedastic (GARCH) specification is a popular nonlinear model that captures the well‐known asymmetric phenomena in financial market data. The switching mechanisms of hysteretic autoregressive GARCH models are different from threshold autoregressive model with GARCH as regime switching may be delayed when the hysteresis variable lies in a hysteresis zone. This paper conducts a Bayesian model comparison among competing models by designing an adaptive Markov chain Monte Carlo sampling scheme. We illustrate the performance of three kinds of criteria by comparing models with fat‐tailed and/or skewed errors: deviance information criteria, Bayesian predictive information, and an asymptotic version of Bayesian predictive information. A simulation study highlights the properties of the three Bayesian criteria and the accuracy as well as their favorable performance as model selection tools. We demonstrate the proposed method in an empirical study of 12 international stock markets, providing evidence to strongly support for both models with skew fat‐tailed innovations. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian Conditional Density Filtering

We propose a conditional density filtering (C-DF) algorithm for efficient online Bayesian inference. C-DF adapts MCMC sampling to the online setting, sampling from approximations to conditional posterior distributions obtained by propagating surrogate conditional sufficient statistics (a function of data and parameter estimates) as new data arrive. These quantities eliminate the need to store or process the entire dataset simultaneously and offer a number of desirable features. Often, these include a reduction in memory requirements and runtime and improved mixing, along with state-of-the-art parameter inference and prediction. These improvements are demonstrated through several illustrative examples including an application to high dimensional compressed regression. In the cases where dimension of the model parameter does not grow with time, we also establish sufficient conditions under which C-DF samples converge to the target posterior distribution asymptotically as sampling proceeds and more data arrive. Supplementary materials of C-DF are available online.     

Modelling non-stationary dynamic gene regulatory processes with the BGM model

Recently, a Bayesian network model for inferring non-stationary regulatory processes from gene expression time series has been proposed. The Bayesian Gaussian Mixture (BGM) Bayesian network model divides the data into disjunct compartments (data subsets) by a free allocation model, and infers network structures, which are kept fixed for all compartments. Fixing the network structure allows for some information sharing among compartments, and each compartment is modelled separately and independently with the Gaussian BGe scoring metric for Bayesian networks. The BGM model can equally be applied to both static (steady-state) and dynamic (time series) gene expression data. However, it is this flexibility that renders its application to time series data suboptimal. To improve the performance of the BGM model on time series data we propose a revised approach in which the free allocation of data points is replaced by a changepoint process so as to take the temporal structure into account. The practical inference follows the Bayesian paradigm and approximately samples the network, the number of compartments and the changepoint locations from the posterior distribution with Markov chain Monte Carlo (MCMC). Our empirical results show that the proposed modification leads to a more efficient inference tool for analysing gene expression time series.     

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.     

状态转移框架下跨地区、跨行业资产配置模型及实证

利用Markov状态转移模型捕捉金融资产收益率序列的非线性、动态的结构性变化,考虑不同市场状态下资金在地区板块、行业板块间流动导致的板块轮动效应,构建基于状态转移的跨地区、跨行业资产配置模型。在此基础上,对市场状态和地区、行业板块轮动效应对资产配置的影响进行细致分析。研究发现:中国股票市场存在明显的动态结构性变化,可以分为熊市状态和牛市状态,两种市场状态下最优资产配置结构不同。结果表明,状态转移框架下的跨地区和跨行业资产配置能够刻画非对称市场状态下资产的收益和风险特征,分散非系统性风险的同时降低市场风险,提高投资者的收益,可以为投资者决策提供有价值的参考。     

Arc refractor methods for adaptive importance sampling on large Bayesian networks under evidential reasoning

Approximate Bayesian inference by importance sampling derives probabilistic statements from a Bayesian network, an essential part of evidential reasoning with the network and an important aspect of many Bayesian methods. A critical problem in importance sampling on Bayesian networks is the selection of a good importance function to sample a network’s prior and posterior probability distribution. The initially optimal importance functions eventually start deviating from the optimal function when sampling a network’s posterior distribution given evidence, even when adaptive methods are used that adjust an importance function to the evidence by learning. In this article we propose a new family of Refractor Importance Sampling (RIS) algorithms for adaptive importance sampling under evidential reasoning. RIS applies “arc refractors” to a Bayesian network by adding new arcs and refining the conditional probability tables. The goal of RIS is to optimize the importance function for the posterior distribution and reduce the error variance of sampling. Our experimental results show a significant improvement of RIS over state-of-the-art adaptive importance sampling algorithms.     

Local non-stationarity test in mean for Markov switching GARCH models: an approximate Bayesian approach

In order to exploit mean-reverting behavior among the price differential between two markets, one can use unit root tests to determine which pairs of assets appear to exhibit mean-reverting behavior. Since nonlinear mean reversion shares the same meaning as local stationarity, this paper proposes a Bayesian hypothesis testing to detect the presence of a local unit root in the mean equation using Markov switching GARCH models. This model incorporates a fat-tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. To implement the test, we propose a numerical approximation of the marginal likelihoods to posterior odds by using an adaptive Markov Chain Monte Carlo scheme. Our simulation study demonstrates that the approximate Bayesian test performs properly. The proposed method utilizes the daily basis between the FTSE 100 Index and Index Futures as an illustration.     

Conditionally heteroscedastic factorial HMMs for time series in finance

In this article, we develop a new approach within the framework of asset pricing models that incorporates two key features of the latent volatility: co‐movement among conditionally heteroscedastic financial returns and switching between different unobservable regimes. By combining latent factor models with hidden Markov chain models we derive a dynamical local model for segmentation and prediction of multivariate conditionally heteroscedastic financial time series. We concentrate more precisely on situations where the factor variances are modelled by univariate generalized quadratic autoregressive conditionally heteroscedastic processes. The expectation maximization algorithm that we have developed for the maximum likelihood estimation is based on a quasi‐optimal switching Kalman filter approach combined with a generalized pseudo‐Bayesian approximation, which yield inferences about the unobservable path of the common factors, their variances and the latent variable of the state process. Extensive Monte Carlo simulations and preliminary experiments obtained with daily foreign exchange rate returns of eight currencies show promising results. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic asset allocation for varied financial markets under regime switching framework

Asset allocation among diverse financial markets is essential for investors especially under situations such as the financial crisis of 2008. Portfolio optimization is the most developed method to examine the optimal decision for asset allocation. We employ the hidden Markov model to identify regimes in varied financial markets; a regime switching model gives multiple distributions and this information can convert the static mean–variance model into an optimization problem under uncertainty, which is the case for unobservable market regimes. We construct a stochastic program to optimize portfolios under the regime switching framework and use scenario generation to mathematically formulate the optimization problem. In addition, we build a simple example for a pension fund and examine the behavior of the optimal solution over time by using a rolling-horizon simulation. We conclude that the regime information helps portfolios avoid risk during left-tail events.     

NHPP models for categorized software defects

We develop NHPP models to characterize categorized event data, with application to modelling the discovery process for categorized software defects. Conditioning on the total number of defects, multivariate models are proposed for modelling the defects by type. A latent vector autoregressive structure is used to characterize dependencies among the different types. We show how Bayesian inference can be achieved via MCMC procedures, with a posterior prediction‐based L‐measure used for model selection. The results are illustrated for defects of different types found during the System Test phase of a large operating system software development project. Copyright © 2005 John Wiley & Sons, Ltd.     

A model of speculative behaviour with a strange attractor

An asset pricing model for a speculative financial market with fundamentalists and chartists is analysed. The model explains bursts of volatility in financial markets, which are not well explained by the traditional finance paradigms. Speculative bubbles arise as a complex non-linear dynamic phenomenon brought about naturally by the dynamic interaction of heterogeneous market participants. Depending on the time lag in the formation of chartists' expectations, the system evolves through several dynamic regimes, finishing in a strange attractor. Chaos provides a self-sustained motion around the rationally expected equilibrium that corresponds to a speculative bubble. In order to explain the role of Chartism, chaotic motion is a very interesting theoretical feature for a speculative financial market model. It provides a complex non-linear dynamic behaviour around the Walrasian equilibrium price produced by deterministic interactions between fundamentalists and chartists. This model could be a link between two opposite views over the behaviour of financial markets: the theorist's literature view that claims the random motion of asset prices, and the chartist's position extensively adopted by market professionals.     

Bayesian inference for the power law process

The power law process has been used to model reliability growth, software reliability and the failure times of repairable systems. This article reviews and further develops Bayesian inference for such a process. The Bayesian approach provides a unified methodology for dealing with both time and failure truncated data. As well as looking at the posterior densities of the parameters of the power law process, inference for the expected number of failures and the probability of no failures in some given time interval is discussed. Aspects of the prediction problem are examined. The results are illustrated with two data examples.