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.     

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al., 2016a, Li et al., 2016b. We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.     

Bayesian stochastic frontier analysis using WinBUGS

Markov chain Monte Carlo (MCMC) methods have become a ubiquitous tool in Bayesian analysis. This paper implements MCMC methods for Bayesian analysis of stochastic frontier models using the WinBUGS package, a freely available software. General code for cross-sectional and panel data are presented and various ways of summarizing posterior inference are discussed. Several examples illustrate that analyses with models of genuine practical interest can be performed straightforwardly and model changes are easily implemented. Although WinBUGS may not be that efficient for more complicated models, it does make Bayesian inference with stochastic frontier models easily accessible for applied researchers and its generic structure allows for a lot of flexibility in model specification.      

Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures

During the last years, graphical models have become a popular tool to represent dependencies among variables in many scientific areas. Typically, the objective is to discover dependence relationships that can be represented through a directed acyclic graph (DAG). The set of all conditional independencies encoded by a DAG determines its Markov property. In general, DAGs encoding the same conditional independencies are not distinguishable from observational data and can be collected into equivalence classes, each one represented by a chain graph called essential graph (EG). However, both the DAG and EG space grow super exponentially in the number of variables, and so, graph structural learning requires the adoption of Markov chain Monte Carlo (MCMC) techniques. In this paper, we review some recent results on Bayesian model selection of Gaussian DAG models under a unified framework. These results are based on closed-form expressions for the marginal likelihood of a DAG and EG structure, which is obtained from a few suitable assumptions on the prior for model parameters. We then introduce a general MCMC scheme that can be adopted both for model selection of DAGs and EGs together with a couple of applications on real data sets.     

What are the advantages of MCMC based inference in latent variable models?

Recent developments in Markov chain Monte Carlo [MCMC] methods have increased the popularity of Bayesian inference in many fields of research in economics, such as marketing research and financial econometrics. Gibbs sampling in combination with data augmentation allows inference in statistical/econometric models with many unobserved variables. The likelihood functions of these models may contain many integrals, which often makes a standard classical analysis difficult or even unfeasible. The advantage of the Bayesian approach using MCMC is that one only has to consider the likelihood function conditional on the unobserved variables. In many cases this implies that Bayesian parameter estimation is faster than classical maximum likelihood estimation. In this paper we illustrate the computational advantages of Bayesian estimation using MCMC in several popular latent variable models.     

Bayes model averaging of cyclical decompositions in economic time series

A flexible decomposition of a time series into stochastic cycles under possible non‐stationarity is specified, providing both a useful data analysis tool and a very wide model class. A Bayes procedure using Markov Chain Monte Carlo (MCMC) is introduced with a model averaging approach which explicitly deals with the uncertainty on the appropriate number of cycles. The convergence of the MCMC method is substantially accelerated through a convenient reparametrization based on a hierarchical structure of variances in a state space model. The model and corresponding inferential procedure are applied to simulated data and to cyclical economic time series like US industrial production and unemployment. We derive the implied posterior distributions of model parameters and some relevant functions thereof, shedding light on several key features of economic time series. Copyright © 2006 John Wiley & Sons, Ltd.     

Stochastic model specification search for Gaussian and partial non-Gaussian state space models

Model specification for state space models is a difficult task as one has to decide which components to include in the model and to specify whether these components are fixed or time-varying. To this aim a new model space MCMC method is developed in this paper. It is based on extending the Bayesian variable selection approach which is usually applied to variable selection in regression models to state space models. For non-Gaussian state space models stochastic model search MCMC makes use of auxiliary mixture sampling. We focus on structural time series models including seasonal components, trend or intervention. The method is applied to various well-known time series.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Stick-breaking autoregressive processes

This paper considers the problem of defining a time-dependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stick-breaking form. The processes with Poisson-Dirichlet and Dirichlet process marginals are investigated in some detail. We develop a general conditional Markov Chain Monte Carlo (MCMC) method for inference in the wide subclass of these models where the parameters of the marginal stick-breaking process are nondecreasing sequences. We derive a generalised Pólya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. We apply the proposed methods to financial data to develop a semi-parametric stochastic volatility model with a time-varying nonparametric returns distribution. Finally, we present two examples concerning the analysis of regional GDP and its growth.     

Computing marginal likelihoods from a single MCMC output

In this article, we propose new Monte Carlo methods for computing a single marginal likelihood or several marginal likelihoods for the purpose of Bayesian model comparisons. The methods are motivated by Bayesian variable selection, in which the marginal likelihoods for all subset variable models are required to compute. The proposed estimates use only a single Markov chain Monte Carlo (MCMC) output from the joint posterior distribution and it does not require the specific structure or the form of the MCMC sampling algorithm that is used to generate the MCMC sample to be known. The theoretical properties of the proposed method are examined in detail. The applicability and usefulness of the proposed method are demonstrated via ordinal data probit regression models. A real dataset involving ordinal outcomes is used to further illustrate the proposed methodology.     

交互效应面板模型的EM算法和MCMC算法

交互效应面板模型是目前计量经济学前沿研究的热点,有着广阔的应用空间。但是对很多应用者而言,模型内的参数估计是一个非常棘手的问题。通常的Newton-Raphson算法在优化似然函数的过程中,常常会出现优化失败的情况。本文依据EM算法和MCMC算法理论,为应用研究者提供了一套获得参数估计值的流程。计算机上的试验证实两种估计方法都非常稳健可靠,并在很多情况下,差异不是很大。     

The structure of dynamic correlations in multivariate stochastic volatility models

This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures can easily be obtained. Both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord through the use of covariance matrices. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure, and simulated data show that the estimation method works well. Various multivariate conditional volatility and MSV models are compared via simulation, including an evaluation of alternative VaR estimators. The DC MSV model is also estimated using three sets of empirical data, namely Nikkei 225 Index, Hang Seng Index and Straits Times Index returns, and significant dynamic correlations are found. The Dynamic Conditional Correlation (DCC) model is also estimated, and is found to be far less sensitive to the covariation in the shocks to the indexes. The correlation process for the DCC model also appears to have a unit root, and hence constant conditional correlations in the long run. In contrast, the estimates arising from the DC MSV model indicate that the dynamic correlation process is stationary.     

Forecasting the Value-at-Risk of REITs using realized volatility jump models

This paper examines jump risk in the time series of Real Estate Investment Trusts (REITs). Using high-frequency index-level and firm-level data, the econometric model in this paper integrates jumps into the volatility forecast by estimating jump augmented Heterogeneous Autoregressive (HAR) models of realized volatility. To assess the information value of these specifications, their forecasting accuracies for generating one-step ahead daily Value-at-Risk are also compared with other VaR specifications, including those generated from historical returns, bootstrap technique, and severity loss distribution.     

Comparison of Sampling Schemes for Dynamic Linear Models

Hyperparameter estimation in dynamic linear models leads to inference that is not available analytically. Recently, the most common approach is through MCMC approximations. A number of sampling schemes that have been proposed in the literature are compared. They basically differ in their blocking structure. In this paper, comparison between the most common schemes is performed in terms of different efficiency criteria, including efficiency ratio and processing time. A sample of time series was simulated to reflect different relevant features such as series length and system volatility.     

Generalized smooth finite mixtures

We propose a general class of models and a unified Bayesian inference methodology for flexibly estimating the density of a response variable conditional on a possibly high-dimensional set of covariates. Our model is a finite mixture of component models with covariate-dependent mixing weights. The component densities can belong to any parametric family, with each model parameter being a deterministic function of covariates through a link function. Our MCMC methodology allows for Bayesian variable selection among the covariates in the mixture components and in the mixing weights. The model’s parameterization and variable selection prior are chosen to prevent overfitting. We use simulated and real data sets to illustrate the methodology.     

Misspecification and Expectations Correction in New Keynesian DSGE Models

This paper focuses on the dynamic misspecification that characterizes the class of small‐scale New Keynesian models currently used in monetary and business cycle analysis, and provides a remedy for the typical difficulties these models have in accounting for the rich contemporaneous and dynamic correlation structure of the data. We suggest using a statistical model for the data as a device through which it is possible to adapt the econometric specification of the New Keynesian model such that the risk of omitting important propagation mechanisms is kept under control. A pseudo‐structural form is built from the baseline system of Euler equations by forcing the state vector of the system to have the same dimension as the state vector characterizing the statistical model. The pseudo‐structural form gives rise to a set of cross‐equation restrictions that do not penalize the autocorrelation structure and persistence of the data. Standard estimation and evaluation methods can be used. We provide an empirical illustration based on USA quarterly data and a small‐scale monetary New Keynesian model.     

On leverage in a stochastic volatility model

This paper is concerned with the specification for modelling financial leverage effect in the context of stochastic volatility (SV) models. Two alternative specifications co-exist in the literature. One is the Euler approximation to the well-known continuous time SV model with leverage effect and the other is the discrete time SV model of Jacquier et al. (J. Econometrics 122 (2004) 185). Using a Gaussian nonlinear state space form with uncorrelated measurement and transition errors, I show that it is easy to interpret the leverage effect in the conventional model whereas it is not clear how to obtain and interpret the leverage effect in the model of Jacquier et al. Empirical comparisons of these two models via Bayesian Markov chain Monte Carlo (MCMC) methods further reveal that the specification of Jacquier et al. is inferior. Simulation experiments are conducted to study the sampling properties of Bayes MCMC for the conventional model.     

Mutual excitation in Eurozone sovereign CDS

We study self- and cross-excitation of shocks in the Eurozone sovereign CDS market. We adopt a multivariate setting with credit default intensities driven by mutually exciting jump processes, to capture the salient features observed in the data, in particular, the clustering of high default probabilities both in time (over days) and in space (across countries). The feedback between jump events and the intensity of these jumps is the key element of the model. We derive closed-form formulae for CDS prices, and estimate the model by matching theoretical prices to their empirical counterparts. We find evidence of self-excitation and asymmetric cross-excitation. Using impulse-response analysis, we assess the impact of shocks and a potential policy intervention not just on a single country under scrutiny but also, through the effect on cross-excitation risk which generates systemic sovereign risk, on other interconnected countries.     

A discrete-time model for daily S & P500 returns and realized variations: Jumps and leverage effects

We develop an empirically highly accurate discrete-time daily stochastic volatility model that explicitly distinguishes between the jump and continuous-time components of price movements using nonparametric realized variation and Bipower variation measures constructed from high-frequency intraday data. The model setup allows us to directly assess the structural inter-dependencies among the shocks to returns and the two different volatility components. The model estimates suggest that the leverage effect, or asymmetry between returns and volatility, works primarily through the continuous volatility component. The excellent fit of the model makes it an ideal candidate for an easy-to-implement auxiliary model in the context of indirect estimation of empirically more realistic continuous-time jump diffusion and Lévy-driven stochastic volatility models, effectively incorporating the interdaily dependencies inherent in the high-frequency intraday data.     

Probabilistic forecasting in day-ahead electricity markets: Simulating peak and off-peak prices

In this article we include dependency structures for electricity price forecasting and forecasting evaluation. We work with off-peak and peak time series from the German-Austrian day-ahead price; hence, we analyze bivariate data. We first estimate the mean of the two time series, and then in a second step we estimate the residuals. The mean equation is estimated by ordinary least squares and the elastic net, and the residuals are estimated by maximum likelihood. Our contribution is to include a bivariate jump component in a mean reverting jump diffusion model in the residuals. The models’ forecasts are evaluated with use of four different criteria, including the energy score to measure whether the correlation structure between the time series is properly included. It is observed that the models with bivariate jumps provide better results with the energy score, which means that it is important to consider this structure to properly forecast correlated time series.