A Tutorial on Reversible Jump MCMC with a View toward Applications in QTL-mapping

A tutorial derivation of the reversible jump Markov chain Monte Carlo (MCMC) algorithm is given. Various examples illustrate how reversible jump MCMC is a general framework for Metropolis-Hastings algorithms where the proposal and the target distribution may have densities on spaces of varying dimension. It is finally discussed how reversible jump MCMC can be applied in genetics to compute the posterior distribution of the number, locations, effects, and genotypes of putative quantitative trait loci.     

Bayesian estimators of the lognormal–Pareto composite distribution

In this paper, Bayesian methods with both Jeffreys and conjugate priors for estimating parameters of the lognormal–Pareto composite (LPC) distribution are considered. With Jeffreys prior, the posterior distributions for parameters of interest are derived and their properties are described. The conjugate priors are proposed and the conditional posterior distributions are provided. In addition, simulation studies are performed to obtain the upper percentage points of Kolmogorov–Smirnov and Anderson–Darling test statistics. Furthermore, these statistics are used to compare Bayesian and likelihood estimators. In order to clarify and advance the validity of Bayesian and likelihood estimators of the LPC distribution, well-known Danish fire insurance data-set is reanalyzed.     

Analysis of high dimensional multivariate stochastic volatility models

This paper is concerned with the Bayesian estimation and comparison of flexible, high dimensional multivariate time series models with time varying correlations. The model proposed and considered here combines features of the classical factor model with that of the heavy tailed univariate stochastic volatility model. A unified analysis of the model, and its special cases, is developed that encompasses estimation, filtering and model choice. The centerpieces of the estimation algorithm (which relies on MCMC methods) are: (1) a reduced blocking scheme for sampling the free elements of the loading matrix and the factors and (2) a special method for sampling the parameters of the univariate SV process. The resulting algorithm is scalable in terms of series and factors and simulation-efficient. Methods for estimating the log-likelihood function and the filtered values of the time-varying volatilities and correlations are also provided. The performance and effectiveness of the inferential methods are extensively tested using simulated data where models up to 50 dimensions and 688 parameters are fit and studied. The performance of our model, in relation to various multivariate GARCH models, is also evaluated using a real data set of weekly returns on a set of 10 international stock indices. We consider the performance along two dimensions: the ability to correctly estimate the conditional covariance matrix of future returns and the unconditional and conditional coverage of the 5% and 1% value-at-risk (VaR) measures of four pre-defined portfolios.     

带未知异方差广义空间模型的有效估计

空间单元大小以及其它的经济特征上的差异,常常会导致空间异方差问题。本文给出了广义空间模型异方差问题的三种不同估计方法。第一种方法是将异方差形式参数化,来克服自由度的不足,使用ML估计进行实现。而针对异方差形式未知时,分别采用了基于2SLS的迭代GMM估计和更加直接的MCMC抽样方法加以解决,特别是MCMC方法表现得更加优美。蒙特卡罗模拟表明,给定异方差形式条件下, ML估计通过异方差参数化的方法依然可以获得较好的估计效果。而异方差形式未知的情况下,另外两种方法随着样本数的增大时也可以与ML的估计结果趋于一致。     

Estimation of risk contributions with MCMC

Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations. In this study, we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from the conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has consistency and asymptotic normality, and is widely applicable to various risk models having a joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models, even those having high-dimensional (≈500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error when compared with existing estimators.     

Time varying VARs with inequality restrictions

In many applications involving time-varying parameter VARs, it is desirable to restrict the VAR coefficients at each point in time to be non-explosive. This is an example of a problem where inequality restrictions are imposed on states in a state space model. In this paper, we describe how existing MCMC algorithms for imposing such inequality restrictions can work poorly (or not at all) and suggest alternative algorithms which exhibit better performance. Furthermore, we show that previous algorithms involve an approximation relating to a key prior integrating constant. Our algorithms are exact, not involving this approximation. In an application involving a commonly used U.S. data set, we present evidence that the algorithms proposed in this paper work well.     

非参数空间滞后模型的贝叶斯估计

本文采用Bayes方法对非参数空间滞后模型进行全面分析,包括参数的估计以及用自由节点样条来拟合未知联系函数。所建议的Bayes方法通过逆跳Markov chain Monte carlo算法（RJMCMC）来实现。在进行贝叶斯分析时,对样条系数与误差方差选取共轭的正态—逆伽玛先验分布,进而获得其他未知量的边际后验分布;另外,文章还设计了一个简单但一般的随机游动Metropolis抽样器,以方便从空间权重因子的条件后验分布中进行抽样。最后应用所建议的方法进行数值模拟。     

Parameter instability,stochastic volatility and estimation based on simulated likelihood: Evidence from the crude oil market

Stochastic volatility models with fixed parameters can be too restrictive for time-series analysis due to instability in the parameters that govern conditional volatility dynamics. We incorporate time-variation in the model parameters for the plain stochastic volatility model as well its extensions with: Leverage, volatility feedback effects and heavy-tailed distributed innovations. With regards to estimation, we rely on one recently discovered result, namely, that when an unbiasedly simulated estimated likelihood (available for example through a particle filter) is used inside a Metropolis-Hastings routine then the estimation error makes no difference to the equilibrium distribution of the algorithm, the posterior distribution. This in turn provides an off-the-shelf technique to estimate complex models. We examine the performance of this technique on simulated and crude oil returns from 1987 to 2016. We find that (i): There is clear evidence of time-variation in the model parameters, (ii): Time-varying parameter volatility models with leverage/Student's t-distributed innovations perform best, (iii): The timing of parameter changes align very well with events such as market turmoils and financial crises.