Bayesian Analysis of DSGE Models—Rejoinder

We would like to thank all the discussants for their stimulating comments. While our article to a large extent reviews current practice of Bayesian analysis of Dynamic Stochastic General Equilibrium (DSGE) models the discussants provide many ideas to improve upon the current practice, thereby outlining a research agenda for the years to come. In our rejoinder we will briefly revisit some of the issues that were raised.     

Bayesian Analysis of DSGE Models

This paper reviews Bayesian methods that have been developed in recent years to estimate and evaluate dynamic stochastic general equilibrium (DSGE) models. We consider the estimation of linearized DSGE models, the evaluation of models based on Bayesian model checking, posterior odds comparisons, and comparisons to vector autoregressions, as well as the non-linear estimation based on a second-order accurate model solution. These methods are applied to data generated from correctly specified and misspecified linearized DSGE models and a DSGE model that was solved with a second-order perturbation method.     

DSGE Models with Student-t Errors

This paper deals with Dynamic Stochastic General Equilibrium (DSGE) models under a multivariate student-t distribution for the structural shocks. Based on the solution algorithm of Klein (2000) and the gamma-normal representation of the t-distribution, the TaRB-MH algorithm of Chib and Ramamurthy (2010 Chib , S. , Ramamurthy , S. ( 2010 ). Tailored randomized block MCMC methods with application to DSGE models . Journal of Econometrics 108 : 19 – 38 .[Crossref], [Web of Science ®] , [Google Scholar]) is used to estimate the model. A technique for estimating the marginal likelihood of the DSGE student-t model is also provided. The methodologies are illustrated first with simulated data and then with the DSGE model of Ireland (2004 Ireland , P. N. ( 2004 ). Technology shocks in the new keynesian model . Review of Economics and Statistics 86 ( 4 ): 923 – 936 .[Crossref], [Web of Science ®] , [Google Scholar]) where the results support the t-error model in relation to the Gaussian model.     

Bayesian Analysis of DSGE Models by S. An and F. Schorfheide

The paper that An and Schorfheide have written is an excellent piece of work and will become a useful reference for teaching and consultation purposes. The paper discusses in an articulate and convincing manner almost everything that one could think of covering in such a review. This makes the task of the commentator difficult. Nevertheless, I will attempt to add few insights on three issues which, in my opinion, play an important role in applied work and in the interpretation of the estimation result. In particular, I will discuss a) the sensitivity of posterior distributions to prior spreads; b) the effects of model misspecification and an approach to model respecification; c) parameter identification and its consequences for posterior inference.     

Bayesian Analysis of DSGE Models by S. An and F. Schorfheide

The paper that An and Schorfheide have written is an excellent piece of work and will become a useful reference for teaching and consultation purposes. The paper discusses in an articulate and convincing manner almost everything that one could think of covering in such a review. This makes the task of the commentator difficult. Nevertheless, I will attempt to add few insights on three issues which, in my opinion, play an important role in applied work and in the interpretation of the estimation result. In particular, I will discuss a) the sensitivity of posterior distributions to prior spreads; b) the effects of model misspecification and an approach to model respecification; c) parameter identification and its consequences for posterior inference.     

情绪冲击、资产泡沫与经济波动:基于贝叶斯DSGE的分析

本文在Smets和Wouters(2003)、Christiano等(2005)模型基础上,引入驱动股价泡沫的情绪冲击,构建了情绪冲击通过资产价格渠道影响经济波动的动态随机一般均衡模型,并采用我国2000-2016年的季度数据对模型进行贝叶斯估计。研究表明,由于企业面临融资约束,正向情绪冲击带来股价泡沫的上升起到了放松信贷约束的作用,因而企业投资增加,进而触发一系列经济变量的顺周期波动。情绪冲击能够解释我国股票价格波动的552%以及顺周期性;劳动供给冲击、技术冲击、投资专有冲击、金融冲击都是我国经济波动的来源,尽管其对产出、消费、投资、劳动时间和股票价格波动的贡献存在异质性。     

基于多重尝试Metropolis算法的非线性DSGE模型贝叶斯推断

本文提出一种改进的多重尝试Metropolis算法,用于非线性动态随机一般均衡模型的贝叶斯参数估计和模型选择。多重尝试策略通过每次迭代抽取多个尝试点的方法来提高算法的混合速率,新方法中提出使用近似的方法提高计算速度,并通过接收概率调整偏差。数值实验表明新方法在相同的计算时间内具有更高的估计效率。最后,本文比较了具有不同货币政策设定的模型对中国经济数据的拟合效果,发现中国数据更加支持具有时变通胀目标的模型。     

Forecasting Performance of an Open Economy DSGE Model

This paper analyzes the forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model, estimated with Bayesian methods, for the Euro area during 1994Q1-2002Q4. We compare the DSGE model and a few variants of this model to various reduced-form forecasting models such as vector autoregressions (VARs) and vector error correction models (VECM), estimated both by maximum likelihood and two different Bayesian approaches, and traditional benchmark models, e.g., the random walk. The accuracy of point forecasts, interval forecasts and the predictive distribution as a whole are assessed in an out-of-sample rolling event evaluation using several univariate and multivariate measures. The results show that the open economy DSGE model compares well with more empirical models and thus that the tension between rigor and fit in older generations of DSGE models is no longer present. We also critically examine the role of Bayesian model probabilities and other frequently used low-dimensional summaries, e.g., the log determinant statistic, as measures of overall forecasting performance.     

Forecasting Performance of an Open Economy DSGE Model

This paper analyzes the forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model, estimated with Bayesian methods, for the Euro area during 1994Q1–2002Q4. We compare the DSGE model and a few variants of this model to various reduced-form forecasting models such as vector autoregressions (VARs) and vector error correction models (VECM), estimated both by maximum likelihood and two different Bayesian approaches, and traditional benchmark models, e.g., the random walk. The accuracy of point forecasts, interval forecasts and the predictive distribution as a whole are assessed in an out-of-sample rolling event evaluation using several univariate and multivariate measures. The results show that the open economy DSGE model compares well with more empirical models and thus that the tension between rigor and fit in older generations of DSGE models is no longer present. We also critically examine the role of Bayesian model probabilities and other frequently used low-dimensional summaries, e.g., the log determinant statistic, as measures of overall forecasting performance.     

财政政策DSGE模型中国化构建及其应用

事前模拟经济对财政政策变化的反应是检验政策效果的重要手段。本文通过国外经典模型中国化改进,并引入系统财政规则,构建财政政策DSGE模型。在有效税率估算校准与参数贝叶斯估计基础上,给出政策模拟检验应用示例。发现税率冲击效应模拟是税制改革实验的有效方法,资本税率可作为经济结构调整的政策工具,以及当前增加政府支出拉动增长作用微弱等结论。该研究也可为我国DSGE模型研究提供参考。     

Bayesian blinded sample size re-estimation

Information before unblinding regarding the success of confirmatory clinical trials is highly uncertain. Current techniques using point estimates of auxiliary parameters for estimating expected blinded sample size: (i) fail to describe the range of likely sample sizes obtained after the anticipated data are observed, and (ii) fail to adjust to the changing patient population. Sequential MCMC-based algorithms are implemented for purposes of sample size adjustments. The uncertainty arising from clinical trials is characterized by filtering later auxiliary parameters through their earlier counterparts and employing posterior distributions to estimate sample size and power. The use of approximate expected power estimates to determine the required additional sample size are closely related to techniques employing Simple Adjustments or the EM algorithm. By contrast with these, our proposed methodology provides intervals for the expected sample size using the posterior distribution of auxiliary parameters. Future decisions about additional subjects are better informed due to our ability to account for subject response heterogeneity over time. We apply the proposed methodologies to a depression trial. Our proposed blinded procedures should be considered for most studies due to ease of implementation.     

A comparison of Bayesian model selection based on MCMC with an application to GARCH-type models

This paper presents a comprehensive review and comparison of five computational methods for Bayesian model selection, based on MCMC simulations from posterior model parameter distributions. We apply these methods to a well-known and important class of models in financial time series analysis, namely GARCH and GARCH-t models for conditional return distributions (assuming normal and t-distributions). We compare their performance with the more common maximum likelihood-based model selection for simulated and real market data. All five MCMC methods proved reliable in the simulation study, although differing in their computational demands. Results on simulated data also show that for large degrees of freedom (where the t-distribution becomes more similar to a normal one), Bayesian model selection results in better decisions in favor of the true model than maximum likelihood. Results on market data show the instability of the harmonic mean estimator and reliability of the advanced model selection methods.     

空间滞后模型的贝叶斯估计

本文采用Bayes方法对空间滞后模型进行全面分析。在构建模型的贝叶斯框架时,对模型系数与误差方差分别选取正态先验分布和逆伽玛先验分布,这样以便获得参数的联合后验分布和条件后验分布。在抽样估计时,文章主要使用MCMC方法,同时还设计了一个简单随机游动Metropolis抽样器,以方便从空间权重因子系数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。     

Bayesian testing of restrictions on vector autoregressive models

In this study, we propose a prior on restricted Vector Autoregressive (VAR) models. The prior setting permits efficient Markov Chain Monte Carlo (MCMC) sampling from the posterior of the VAR parameters and estimation of the Bayes factor. Numerical simulations show that when the sample size is small, the Bayes factor is more effective in selecting the correct model than the commonly used Schwarz criterion. We conduct Bayesian hypothesis testing of VAR models on the macroeconomic, state-, and sector-specific effects of employment growth.     

Bayesian Analysis of a Growth Curve Model with a General Autoregressive Covariance Structure

In this paper we consider from maximum likelihood and Bayesian points of view the generalized growth curve model when the covariance matrix has a Toeplitz structure. This covariance is a generalization of the AR(1) dependence structure. Inferences on the parameters as well as the future values are included. The results are illustrated with several real data sets.     

Bayesian Analysis of DSGE Models—Rejoinder

We would like to thank all the discussants for their stimulating comments. While our article to a large extent reviews current practice of Bayesian analysis of Dynamic Stochastic General Equilibrium (DSGE) models the discussants provide many ideas to improve upon the current practice, thereby outlining a research agenda for the years to come. In our rejoinder we will briefly revisit some of the issues that were raised.     

Bayesian Time Series Analysis of Structural Changes in Level and Trend

In this article we consider the problem of detecting changes in level and trend in time series model in which the number of change-points is unknown. The approach of Bayesian stochastic search model selection is introduced to detect the configuration of changes in a time series. The number and positions of change-points are determined by a sequence of change-dependent parameters. The sequence is estimated by its posterior distribution via the maximum a posteriori (MAP) estimation. Markov chain Monte Carlo (MCMC) method is used to estimate posterior distributions of parameters. Some actual data examples including a time series of traffic accidents and two hydrological time series are analyzed.     

Computing Marginal Likelihoods via Posterior Sampling

This article presents a novel and simple approach to the estimation of a marginal likelihood, in a Bayesian context. The estimate is based on a Markov chain output which provides samples from the posterior distribution and an additional latent variable. It is the mean of this latent variable which provides the estimate for the value of the marginal likelihood.     

Higher-order Bayesian Approximations for Pseudo-posterior Distributions

The theory of higher-order asymptotics provides accurate approximations to posterior distributions for a scalar parameter of interest, and to the corresponding tail area, for practical use in Bayesian analysis. The aim of this article is to extend these approximations to pseudo-posterior distributions, e.g., posterior distributions based on a pseudo-likelihood function and a suitable prior, which are proved to be particularly useful when the full likelihood is analytically or computationally infeasible. In particular, from a theoretical point of view, we derive the Laplace approximation for a pseudo-posterior distribution, and for the corresponding tail area, for a scalar parameter of interest, also in the presence of nuisance parameters. From a computational point of view, starting from these higher-order approximations, we discuss the higher-order tail area (HOTA) algorithm useful to approximate marginal posterior distributions, and related quantities. Compared to standard Markov chain Monte Carlo methods, the main advantage of the HOTA algorithm is that it gives independent samples at a negligible computational cost. The relevant computations are illustrated by two examples.     

Objective Bayesian Analysis for Log-logistic Distribution

In this article, Bayesian approach is applied to estimate the parameters of Log-logistic distribution under reference prior and Jeffreys’ prior. The reference prior is derived and it is found that the reference prior is also a second-order matching priors as for the case of any parameter of interest. The Bayesian estimators cannot be obtained in explicit forms. Metropolis within Gibbs sampling algorithm is used to obtain the Bayesian estimators. The Bayesian estimates are compared with the maximum likelihood estimates via simulation study. A real dataset is considered for illustrative purposes.