Steady‐state priors for vector autoregressions

Bayesian priors are often used to restrain the otherwise highly over‐parametrized vector autoregressive (VAR) models. The currently available Bayesian VAR methodology does not allow the user to specify prior beliefs about the unconditional mean, or steady state, of the system. This is unfortunate as the steady state is something that economists usually claim to know relatively well. This paper develops easily implemented methods for analyzing both stationary and cointegrated VARs, in reduced or structural form, with an informative prior on the steady state. We document that prior information on the steady state leads to substantial gains in forecasting accuracy on Swedish macro data. A second example illustrates the use of informative steady‐state priors in a cointegration model of the consumption‐wealth relationship in the USA. Copyright © 2009 John Wiley & Sons, Ltd.     

IDENTIFICATION ISSUES IN LIMITED‐INFORMATION BAYESIAN ANALYSIS OF STRUCTURAL MACROECONOMIC MODELS

The likelihood of the parameters in structural macroeconomic models typically has non‐identification regions over which it is constant. When sufficiently diffuse priors are used, the posterior piles up in such non‐identification regions. Use of informative priors can lead to the opposite, so both can generate spurious inference. We propose priors/posteriors on the structural parameters that are implied by priors/posteriors on the parameters of an embedding reduced‐form model. An example of such a prior is the Jeffreys prior. We use it to conduct Bayesian limited‐information inference on the new Keynesian Phillips curve with a VAR reduced form for US data. Copyright © 2014 John Wiley & Sons, Ltd.     

Identification structure of linear structural models

In this paper identification problems in linear structural models are discussed. A special but practically important form of these problems, the structural identifiability, is defined. Thereafter it is shown that this problem can be studied — without empirical data — by a set system defined on the set of free parameters of a model. This set system is the identification structure, or the matroid, of the model. It gives insight into the dependence of unidentified parameters. The advantages of this approach in research planning and in correcting nonidentifiable models is demonstrated. Finally we introduce the FORTRAN program LISRAN, which is designed to analyze a given linear structural model and to provide its identification structure.     

Data‐Driven Identification Constraints for DSGE Models

We propose imposing data‐driven identification constraints to alleviate the multimodality problem arising in the estimation of poorly identified dynamic stochastic general equilibrium models under non‐informative prior distributions. We also devise an iterative procedure based on the posterior density of the parameters for finding these constraints. An empirical application to the Smets and Wouters ( 2007 ) model demonstrates the properties of the estimation method, and shows how the problem of multimodal posterior distributions caused by parameter redundancy is eliminated by identification constraints. Out‐of‐sample forecast comparisons as well as Bayes factors lend support to the constrained model.     

Mixed Hidden Markov Models for Longitudinal Data: An Overview

In this paper we review statistical methods which combine hidden Markov models (HMMs) and random effects models in a longitudinal setting, leading to the class of so‐called mixed HMMs. This class of models has several interesting features. It deals with the dependence of a response variable on covariates, serial dependence, and unobserved heterogeneity in an HMM framework. It exploits the properties of HMMs, such as the relatively simple dependence structure and the efficient computational procedure, and allows one to handle a variety of real‐world time‐dependent data. We give details of the Expectation‐Maximization algorithm for computing the maximum likelihood estimates of model parameters and we illustrate the method with two real applications describing the relationship between patent counts and research and development expenditures, and between stock and market returns via the Capital Asset Pricing Model.     

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters.     

Fixed and Random Effects in Classical and Bayesian Regression*

This paper proposes a common and tractable framework for analyzing fixed and random effects models, in particular constant‐slope variable‐intercept designs. It is shown that, regardless of whether effects (i) are treated as parameters or as an error term, (ii) are estimated in different stages of a hierarchical model, or whether (iii) correlation between effects and regressors is allowed, when the same prior information on idiosyncratic parameters is introduced into all estimation methods, the resulting common slope estimator is also the same across methods. These results are illustrated using the Grünfeld investment data with different prior distributions. Random effects estimates are shown to be more efficient than fixed effects estimates. This efficiency gain, however, comes at the cost of neglecting information obtained in the computation of the prior unknown variance of idiosyncratic parameters.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.     

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.     

On modeling panels of time series*

This paper reviews research issues in modeling panels of time series. Examples of this type of data are annually observed macroeconomic indicators for all countries in the world, daily returns on the individual stocks listed in the S&P500, and the sales records of all items in a retail store. A panel of time series concerns the case where the cross‐sectional dimension and the time dimension are large. Often, there is no a priori reason to select a few series or to aggregate the series over the cross‐sectional dimension. The use of, for example, a vector autoregression or other types of multivariate models then becomes cumbersome. Panel models and associated estimation techniques are more useful. Due to the large time dimension, one should however incorporate the time‐series features. And, the models should not have too many parameters to facilitate interpretation. This paper discusses representation, estimation and inference of relevant models and discusses recently proposed modeling approaches that explicitly aim to meet these requirements. The paper concludes with some reflections on the usefulness of large data sets. These concern sample selection issues and the notion that more detail also requires more complex models.     

Functional-coefficient cointegration models

This paper studies estimation and inference of functional coefficient cointegration models. The proposed model offers a more flexible structure of cointegration where the value of cointegrating coefficients may be affected by informative covariates and thus may vary over time. The model may be viewed as a stochastic cointegration model and includes the conventional cointegration model as a special case. The proposed new model provides a useful complement to the conventional fixed coefficient cointegration models. Both kernel and local polynomial estimators are investigated. Inference procedures for instability of cointegrating parameters and a test for cointegration are proposed based on the functional-coefficient estimates. Limiting distributions of the estimates and testing statistics are derived.     

Minimax‐Optimal Strength of Statistical Evidence for a Composite Alternative Hypothesis

While the likelihood ratio measures statistical support for an alternative hypothesis about a single parameter value, it is undefined for an alternative hypothesis that is composite in the sense that it corresponds to multiple parameter values. Regarding the parameter of interest as a random variable enables measuring support for a composite alternative hypothesis without requiring the elicitation or estimation of a prior distribution, as described below. In this setting, in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log‐odds. That ideal support may be replaced by any measure of support that, on a per‐observation basis, is asymptotically unbiased as a predictor of the ideal support. Such measures of support are easily interpreted and, if desired, can be combined with any specified or estimated prior probability of the null hypothesis. Two qualifying measures of support are minimax‐optimal. An application to proteomics data indicates that a modification of optimal support computed from data for a single protein can closely approximate the estimated difference in posterior and prior odds that would be available with the data for 20 proteins.     

Semiparametric Bayesian inference in multiple equation models

This paper outlines an approach to Bayesian semiparametric regression in multiple equation models which can be used to carry out inference in seemingly unrelated regressions or simultaneous equations models with nonparametric components. The approach treats the points on each nonparametric regression line as unknown parameters and uses a prior on the degree of smoothness of each line to ensure valid posterior inference despite the fact that the number of parameters is greater than the number of observations. We develop an empirical Bayesian approach that allows us to estimate the prior smoothing hyperparameters from the data. An advantage of our semiparametric model is that it is written as a seemingly unrelated regressions model with independent normal–Wishart prior. Since this model is a common one, textbook results for posterior inference, model comparison, prediction and posterior computation are immediately available. We use this model in an application involving a two‐equation structural model drawn from the labour and returns to schooling literatures. Copyright © 2005 John Wiley & Sons, Ltd.     

Natural conjugate priors for the instrumental variables regression model applied to the Angrist–Krueger data

We propose a natural conjugate prior for the instrumental variables regression model. The prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior. The Jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator. We construct informative priors for the Angrist–Krueger [1991. Does compulsory school attendance affect schooling and earnings? Quarterly Journal of Economics 106, 979–1014] data and show that the marginal posterior of the return on education in the US coincides with the marginal posterior from the Southern region when we use the Jeffreys prior. This result occurs since the instruments are the strongest in the Southern region and the posterior using the Jeffreys prior, identical to maximum likelihood, focusses on the strongest available instruments. We construct informative priors for the other regions that make their posteriors of the return on education similar to that of the US and the Southern region. These priors show the amount of prior information needed to obtain comparable results for all regions.     

基于Logistic模型的物流业上市公司财务预警

从物流业104家A股上市公司中选取了60家上市公司作为样本,分别用提取主成分构建逻辑回归模型和原指标构建逻辑回归模型的方法进行财务预警研究,得出了以下结论:用聚类分析方法判断上司公司是否处于财务困境能减少误差且更具有现实意义,方便投资者做出投资决策;主成分构建的逻辑回归模型和原指标构建的逻辑回归模型预测效果并无显著差异,构建的Logistic模型预测正确性都在80%左右,预测效果良好;在以后研究物流业上市公司财务预警时,必须考虑选取主营业务收入增长率、流动资产周转率、现金债务总额比和现金流量比率四个财务指标。     

Switching generalized autoregressive score copula models with application to systemic risk

Recent financial disasters have emphasized the need to accurately predict extreme financial losses and their consequences for the institutions belonging to a given financial market. The ability of econometric models to predict extreme events strongly relies on their flexibility to account for the highly nonlinear and asymmetric dependence patterns observed in financial time series. In this paper, we develop a new class of flexible copula models where the dependence parameters evolve according to a Markov switching generalized autoregressive score (GAS) dynamics. Maximum likelihood estimation is performed using a two‐step procedure where the second step relies on the expectation–maximization algorithm. The proposed switching GAS copula models are then used to estimate the conditional value at risk and the conditional expected shortfall, measuring the impact on an institution of extreme events affecting another institution or the market. The empirical investigation, conducted on a panel of European regional portfolios, reveals that the proposed model is able to explain and predict the evolution of the systemic risk contributions over the period 1999–2015.     

Are differences in firm size transitory or permanent?

An Erratum for this article has been published in Journal of Applied Econometrics 18(2) 2003, 249 Previous empirical work on corporate growth rates using cross‐section or short‐panel econometric techniques suggests that growth rates are random but that some degree of mean reversion exists. This means that size differences between firms are transitory. Another, more natural way to explore the long‐run distribution of firm sizes is to examine data on the growth of particular firms over long periods of time. Using a sample of 147 UK firms observed continually for more than 30 years, our conclusions are that growth rates are highly variable over time and that differences in growth rates between firms do not persist for very long. Further, firms show no tendency to converge to either a common size or to a pattern of stable size differences over time. These results are compared and contrasted with standard approaches that suggest that firms reach and maintain stable positions in a skewed size distribution. Copyright © 2002 John Wiley & Sons, Ltd.     

Sparse Partial Least Squares in Time Series for Macroeconomic Forecasting

Factor models have been applied extensively for forecasting when high‐dimensional datasets are available. In this case, the number of variables can be very large. For instance, usual dynamic factor models in central banks handle over 100 variables. However, there is a growing body of literature indicating that more variables do not necessarily lead to estimated factors with lower uncertainty or better forecasting results. This paper investigates the usefulness of partial least squares techniques that take into account the variable to be forecast when reducing the dimension of the problem from a large number of variables to a smaller number of factors. We propose different approaches of dynamic sparse partial least squares as a means of improving forecast efficiency by simultaneously taking into account the variable forecast while forming an informative subset of predictors, instead of using all the available ones to extract the factors. We use the well‐known Stock and Watson database to check the forecasting performance of our approach. The proposed dynamic sparse models show good performance in improving efficiency compared to widely used factor methods in macroeconomic forecasting. Copyright © 2014 John Wiley & Sons, Ltd.     

A sequential Monte Carlo approach to inference in multiple‐equation Markov‐switching models

Vector autoregressions with Markov‐switching parameters (MS‐VARs) offer substantial gains in data fit over VARs with constant parameters. However, Bayesian inference for MS‐VARs has remained challenging, impeding their uptake for empirical applications. We show that sequential Monte Carlo (SMC) estimators can accurately estimate MS‐VAR posteriors. Relative to multi‐step, model‐specific MCMC routines, SMC has the advantages of generality, parallelizability, and freedom from reliance on particular analytical relationships between prior and likelihood. We use SMC's flexibility to demonstrate that model selection among MS‐VARs can be highly sensitive to the choice of prior.     

Trends and cycles in economic time series: A Bayesian approach

Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be very informative for policy makers, particularly with regard to the size and direction of the output gap and potential turning points. From the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing Markov chain Monte Carlo methods for both univariate and multivariate models. Applications to US macroeconomic series are presented.