Bayesian Analysis of DSGE Models—Some Comments

Sungbae An and Frank Schorfheide have provided an excellent review of the main elements of Bayesian inference in Dynamic Stochastic General Equilibrium (DSGE) models. Bayesian methods have, for reasons clearly outlined in the paper, a very natural role to play in DSGE analysis, and the appeal of the Bayesian paradigm is indeed strongly evidenced by the flood of empirical applications in the area over the last couple of years. We expect their paper to be the natural starting point for applied economists interested in learning about Bayesian techniques for analyzing DSGE models, and as such the paper is likely to have a strong influence on what will be considered best practice for estimating DSGE models.

The authors have, for good reasons, chosen a stylized six-equation model to present the methodology. We shall use here the large-scale model in Adolfson et al. (2005), henceforth ALLV, to illustrate a few econometric problems which we have found to be especially important as the size of the model increases. The model in ALLV is an open economy extension of the closed economy model in Christiano et al. (2005). It consists of 25 log-linearized equations, which can be written as a state space representation with 60 state variables, many of them unobserved. Fifteen observed unfiltered time series are used to estimate 51 structural parameters. An additional complication compared to the model in An and Schorfheide's paper is that some of the coefficients in the measurement equation are non-linear functions of the structural parameters. The model is currently the main vehicle for policy analysis at Sveriges Riksbank (Central Bank of Sweden) and similar models are being developed in many other policy institutions, which testifies to the model's practical relevance. The version considered here is estimated on Euro area data over the period 1980Q1-2002Q4. We refer to ALLV for details.     

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.     

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.     

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.     

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.     

Bayesian analysis of generalized elliptical semi-parametric models

In this paper, we study the statistical inference based on the Bayesian approach for regression models with the assumption that independent additive errors follow normal, Student-t, slash, contaminated normal, Laplace or symmetric hyperbolic distribution, where both location and dispersion parameters of the response variable distribution include nonparametric additive components approximated by B-splines. This class of models provides a rich set of symmetric distributions for the model error. Some of these distributions have heavier or lighter tails than the normal as well as different levels of kurtosis. In order to draw samples of the posterior distribution of the interest parameters, we propose an efficient Markov Chain Monte Carlo (MCMC) algorithm, which combines Gibbs sampler and Metropolis–Hastings algorithms. The performance of the proposed MCMC algorithm is assessed through simulation experiments. We apply the proposed methodology to a real data set. The proposed methodology is implemented in the R package BayesGESM using the function gesm().     

Bayesian Approach to Multicentre Sparse Data

In a 2 × 2 contingency table, when the sample size is small, there may be a number of cells that contain few or no observations, usually referred to as sparse data. In such cases, a common recommendation in the conventional frequentist methods is adding a small constant to every cell of the observed table to find the estimates of the unknown parameters. However, this approach is based on asymptotic properties of the estimates and may work poorly for small samples. An alternative approach would be to use Bayesian methods in order to provide better insight into the problem of sparse data coupled with fewer centers, which would otherwise be difficult to carry out the analysis. In this article, an attempt has been made to use hierarchical Bayesian model to a multicenter data on the effect of a surgical treatment with standard foot care among leprosy patients with posterior tibial nerve damage which is summarized as seven 2 × 2 tables. Monte Carlo Markov Chain (MCMC) techniques are applied in estimating the parameters of interest under sparse data setup.     

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。     

Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.     

Mixture of Distributions in the Biparametric Exponential Family: A Bayesian Approach

In this article we propose mixture of distributions belonging to the biparametric exponential family, considering joint modeling of the mean and variance (or dispersion) parameters. As special cases we consider mixtures of normal and gamma distributions. A novel Bayesian methodology, using Markov Chain Monte Carlo (MCMC) methods, is proposed to obtain the posterior summaries of interest. We include simulations and real data examples to illustrate de performance of the proposal.     

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。     

The Use of Working Variables in the Bayesian Modeling of Mean and Dispersion Parameters in Generalized Nonlinear Models with Random Effects

This article is aimed at reviewing a novel Bayesian approach to handle inference and estimation in the class of generalized nonlinear models. These models include some of the main techniques of statistical methodology, namely generalized linear models and parametric nonlinear regression. In addition, this proposal extends to methods for the systematic treatment of variation that is not explicitly predicted within the model, through the inclusion of random effects, and takes into account the modeling of dispersion parameters in the class of two-parameter exponential family. The methodology is based on the implementation of a two-stage algorithm that induces a hybrid approach based on numerical methods for approximating the likelihood to a normal density using a Taylor linearization around the values of current parameters in an MCMC routine.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

基于逆跳MCMC的贝叶斯分位自回归模型研究

考虑到传统信息理论方法确定模型存在不足,在贝叶斯理论框架下提出了基于逆跳马尔可夫链蒙特卡罗法确定分位自回归模型阶次的方法。在时间序列服从非对称Laplace分布的条件下,设计了马尔可夫链蒙特卡罗数值计算程序,得到了不同分位数下模型参数的贝叶斯估计值。实证研究表明：基于逆跳马尔可夫链蒙特卡罗法的贝叶斯分位自回归模型能有效地揭示滞后变量对响应变量的位置、尺度和形状的影响。     

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 variable selection in quantile regression with random effects: an application to Municipal Human Development Index

According to the Atlas of Human Development in Brazil, the income dimension of Municipal Human Development Index (MHDI-I) is an indicator that shows the population''s ability in a municipality to ensure a minimum standard of living to provide their basic needs, such as water, food and shelter. In public policy, one of the research objectives is to identify social and economic variables that are associated with this index. Due to the income inequality, evaluate these associations in quantiles, instead of the mean, could be more interest. Thus, in this paper, we develop a Bayesian variable selection in quantile regression models with hierarchical random effects. In particular, we assume a likelihood function based on the Generalized Asymmetric Laplace distribution, and a spike-and-slab prior is used to perform variable selection. The Generalized Asymmetric Laplace distribution is a more general alternative than the Asymmetric Laplace one, which is a common approach used in quantile regression under the Bayesian paradigm. The performance of the proposed method is evaluated via a comprehensive simulation study, and it is applied to the MHDI-I from municipalities located in the state of Rio de Janeiro.