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.     

Steady‐state modeling and macroeconomic forecasting quality

Vector autoregressions (VARs) with informative steady‐state priors are standard forecasting tools in empirical macroeconomics. This study proposes (i) an adaptive hierarchical normal‐gamma prior on steady states, (ii) a time‐varying steady‐state specification which accounts for structural breaks in the unconditional mean, and (iii) a generalization of steady‐state VARs with fat‐tailed and heteroskedastic error terms. Empirical analysis, based on a real‐time dataset of 14 macroeconomic variables, shows that, overall, the hierarchical steady‐state specifications materially improve out‐of‐sample forecasting for forecasting horizons longer than 1 year, while the time‐varying specifications generate superior forecasts for variables with significant changes in their unconditional mean.     

‘Objective’ bayesian unit root tests

Due to weaknesses in traditional tests, a Bayesian approach is developed to investigate whether unit roots exist in macroeconomic time-series. Bayesian posterior odds comparing unit root models to stationary and trend-stationary alternatives are calculated using informative priors. Two classes of reference priors which are informative but require minimal subjective prior input are used. In this sense the Bayesian unit root tests developed here are objective. Bayesian procedures are carried out on the Nelson–Plosser and Shiller data sets as well as on generated data. The conclusion is that the failure of classical procedures to reject the unit root hypothesis is not necessarily proof that a unit root is present with high probability.     

Forecasting using variational Bayesian inference in large vector autoregressions with hierarchical shrinkage

Many recent papers in macroeconomics have used large vector autoregressions (VARs) involving 100 or more dependent variables. With so many parameters to estimate, Bayesian prior shrinkage is vital to achieve reasonable results. Computational concerns currently limit the range of priors used and render difficult the addition of empirically important features such as stochastic volatility to the large VAR. In this paper, we develop variational Bayesian methods for large VARs that overcome the computational hurdle and allow for Bayesian inference in large VARs with a range of hierarchical shrinkage priors and with time-varying volatilities. We demonstrate the computational feasibility and good forecast performance of our methods in an empirical application involving a large quarterly US macroeconomic data set.     

Bayesian interval estimation for the difference of two independent Poisson rates using data subject to under‐reporting

Comparing occurrence rates of events of interest in science, business, and medicine is an important topic. Because count data are often under‐reported, we desire to account for this error in the response when constructing interval estimators. In this article, we derive a Bayesian interval for the difference of two Poisson rates when counts are potentially under‐reported. The under‐reporting causes a lack of identifiability. Here, we use informative priors to construct a credible interval for the difference of two Poisson rate parameters with under‐reported data. We demonstrate the efficacy of our new interval estimates using a real data example. We also investigate the performance of our newly derived Bayesian approach via simulation and examine the impact of various informative priors on the new interval.     

STRATEGIC ASSET ALLOCATION FOR LONG‐TERM INVESTORS: PARAMETER UNCERTAINTY AND PRIOR INFORMATION

We study the effect of parameter uncertainty on the long‐run risk for three asset classes: stocks, bills and bonds. Using a Bayesian vector autoregression with an uninformative prior we find that parameter uncertainty raises the annualized long‐run volatilities of all three asset classes proportionally with the same factor relative to volatilities that are conditional on maximum likelihood parameter estimates. As a result, the horizon effect in optimal asset allocations is much weaker compared to models in which only equity returns are subject to parameter uncertainty. Results are sensitive to alternative informative priors, but generally the term structure of risk for stocks and bonds is relatively flat for investment horizons up to 15 years. Copyright © 2013 John Wiley & Sons, Ltd.     

Forecasting with Medium and Large Bayesian VARS

This paper is motivated by the recent interest in the use of Bayesian VARs for forecasting, even in cases where the number of dependent variables is large. In such cases factor methods have been traditionally used, but recent work using a particular prior suggests that Bayesian VAR methods can forecast better. In this paper, we consider a range of alternative priors which have been used with small VARs, discuss the issues which arise when they are used with medium and large VARs and examine their forecast performance using a US macroeconomic dataset containing 168 variables. We find that Bayesian VARs do tend to forecast better than factor methods and provide an extensive comparison of the strengths and weaknesses of various approaches. Typically, we find that the simple Minnesota prior forecasts well in medium and large VARs, which makes this prior attractive relative to computationally more demanding alternatives. Our empirical results show the importance of using forecast metrics based on the entire predictive density, instead of relying solely on those based on point forecasts. Copyright © 2011 John Wiley & Sons, Ltd.     

To criticize the critics: An objective bayesian analysis of stochastic trends

In two recent articles, Sims (1988) and Sims and Uhlig (1988/1991) question the value of much of the ongoing literature on unit roots and stochastic trends. They characterize the seeds of this literature as ‘sterile ideas’, the application of nonstationary limit theory as ‘wrongheaded and unenlightening’, and the use of classical methods of inference as ‘unreasonable’ and ‘logically unsound’. They advocate in place of classical methods an explicit Bayesian approach to inference that utilizes a flat prior on the autoregressive coefficient. DeJong and Whiteman adopt a related Bayesian approach in a group of papers (1989a,b,c) that seek to re-evaluate the empirical evidence from historical economic time series. Their results appear to be conclusive in turning around the earlier, influential conclusions of Nelson and Plosser (1982) that most aggregate economic time series have stochastic trends. So far these criticisms of unit root econometrics have gone unanswered; the assertions about the impropriety of classical methods and the superiority of flat prior Bayesian methods have been unchallenged; and the empirical re-evaluation of evidence in support of stochastic trends has been left without comment. This paper breaks that silence and offers a new perspective. We challenge the methods, the assertions, and the conclusions of these articles on the Bayesian analysis of unit roots. Our approach is also Bayesian but we employ what are known in the statistical literature as objective ignorance priors in our analysis. These are developed in the paper to accommodate explicitly time series models in which no stationarity assumption is made. Ignorance priors are intended to represent a state of ignorance about the value of a parameter and in many models are very different from flat priors. We demonstrate that in time series models flat priors do not represent ignorance but are actually informative (sic) precisely because they neglect generically available information about how autoregressive coefficients influence observed time series characteristics. Contrary to their apparent intent, flat priors unwittingly bias inferences towards stationary and i.i.d. alternatives where they do represent ignorance, as in the linear regression model. This bias helps to explain the outcome of the simulation experiments in Sims and Uhlig and some of the empirical results of DeJong and Whiteman. Under both flat priors and ignorance priors this paper derives posterior distributions for the parameters in autoregressive models with a deterministic trend and an arbitrary number of lags. Marginal posterior distributions are obtained by using the Laplace approximation for multivariate integrals along the lines suggested by the author (Phillips, 1983) in some earlier work. The bias towards stationary models that arises from the use of flat priors is shown in our simulations to be substantial; and we conclude that it is unacceptably large in models with a fitted deterministic trend, for which the expected posterior probability of a stochastic trend is found to be negligible even though the true data generating mechanism has a unit root. Under ignorance priors, Bayesian inference is shown to accord more closely with the results of classical methods. An interesting outcome of our simulations and our empirical work is the bimodal Bayesian posterior, which demonstrates that Bayesian confidence sets can be disjoint, just like classical confidence intervals that are based on asymptotic theory. The paper concludes with an empirical application of our Bayesian methodology to the Nelson-Plosser series. Seven of the 14 series show evidence of stochastic trends under ignorance priors, whereas under flat priors on the coefficients all but three of the series appear trend stationary. The latter result corresponds closely with the conclusion reached by DeJong and Whiteman (1989b) (based on truncated flat priors). We argue that the DeJong-Whiteman inferences are biased towards trend stationarity through the use of flat priors on the autoregressive coefficients, and that their inferences for some of the series (especially stock prices) are fragile (i.e. not robust) not only to the prior but also to the lag length chosen in the time series specification.     

A Review and Empirical Comparison of Bayesian and Classical Approaches to Inference on Efficiency Levels in Stochastic Frontier Models with Panel Data

This paper appliesa large number of models to three previously-analyzed data sets,and compares the point estimates and confidence intervals fortechnical efficiency levels. Classical procedures include multiplecomparisons with the best, based on the fixed effects estimates;a univariate version, marginal comparisons with the best; bootstrappingof the fixed effects estimates; and maximum likelihood givena distributional assumption. Bayesian procedures include a Bayesianversion of the fixed effects model, and various Bayesian modelswith informative priors for efficiencies. We find that fixedeffects models generally perform poorly; there is a large payoffto distributional assumptions for efficiencies. We do not findmuch difference between Bayesian and classical procedures, inthe sense that the classical MLE based on a distributional assumptionfor efficiencies gives results that are rather similar to a Bayesiananalysis with the corresponding prior.     

Bayesian Model Selection with an Uninformative Prior*

Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill‐defined. Using careful consideration of the parameter of interest in cointegration analysis and a re‐specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283–306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with ‘ignorance’ priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model.     

Forecasting with vector autoregressive (VAR) models subject to business cycle restrictions

In the last decade VAR models have become a widely-used tool for forecasting macroeconomic time series. To improve the out-of-sample forecasting accuracy of these models, Bayesian random-walk prior restrictions are often imposed on VAR model parameters. This paper focuses on whether placing an alternative type of restriction on the parameters of unrestricted VAR models improves the out-of-sample forecasting performance of these models. The type of restriction analyzed here is based on the business cycle characteristics of U.S. macroeconomic data, and in particular, requires that the dynamic behavior of the restricted VAR model mimic the business cycle characteristics of historical data. The question posed in this paper is: would a VAR model, estimated subject to the restriction that the cyclical characteristics of simulated data from the model “match up” with the business cycle characteristics of U.S. data, generate more accurate out-of-sample forecasts than unrestricted or Bayesian VAR models?     

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.     

Empirical factor demands and flexible functional forms: a bayesian approach

In this paper we compare classical econometrics, calibration and Bayesian inference in the context of the empirical analysis of factor demands. Our application is based on a popular flexible functional form for the firm's cost function, namely Diewert's Generalized Leontief function, and uses the well-known Berndt and Wood 1947–1971 KLEM data on the US manufacturing sector. We illustrate how the Gibbs sampling methodology can be easily used to calibrate parameter values and elasticities on the basis of previous knowledge from alternative studies on the same data, but with different functional forms. We rely on a system of mixed non-informative diffuse priors for some key parameters and informative tight priors for others. Within the Gibbs sampler, we employ rejection sampling to incorporate parameter restrictions, which are suggested by economic theory but in general rejected by economic data. Our results show that values of those parameters that relate to non-informative priors are almost equal to the standard SUR estimates, whereas differences come out for those parameters to which we have assigned informative priors. Moreover, discrepancies can be appreciated in some crucial parameter estimates obtained with or without rejection sampling.     

Bayesian doubly adaptive elastic-net Lasso for VAR shrinkage

We develop a novel Bayesian doubly adaptive elastic-net Lasso (DAELasso) approach for VAR shrinkage. DAELasso achieves variable selection and coefficient shrinkage in a data-based manner. It deals constructively with explanatory variables which tend to be highly collinear by encouraging the grouping effect. In addition, it also allows for different degrees of shrinkage for different coefficients. Rewriting the multivariate Laplace distribution as a scale mixture, we establish closed-form conditional posteriors that can be drawn from a Gibbs sampler. An empirical analysis shows that the forecast results produced by DAELasso and its variants are comparable to those from other popular Bayesian methods, which provides further evidence that the forecast performances of large and medium sized Bayesian VARs are relatively robust to prior choices, and, in practice, simple Minnesota types of priors can be more attractive than their complex and well-designed alternatives.     

A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model

Computationally efficient methods for Bayesian analysis of seemingly unrelated regression (SUR) models are described and applied that involve the use of a direct Monte Carlo (DMC) approach to calculate Bayesian estimation and prediction results using diffuse or informative priors. This DMC approach is employed to compute Bayesian marginal posterior densities, moments, intervals and other quantities, using data simulated from known models and also using data from an empirical example involving firms’ sales. The results obtained by the DMC approach are compared to those yielded by the use of a Markov Chain Monte Carlo (MCMC) approach. It is concluded from these comparisons that the DMC approach is worthwhile and applicable to many SUR and other problems.     

CO‐INTEGRATING VAR MODELS AND ECONOMIC POLICY

This paper surveys some relevant contributions to the economic literature on co‐integrating vector autoregressive (VAR) models [vector error correction mechanisms (VECMs)], emphasizing their usefulness for economic policy. It further discusses some theoretical aspects that are necessary for a complete understanding of their potential. The theoretical introduction of the co‐integrating VAR model is followed by an illustration of its applications to monetary policy, fiscal policy and exchanges rates as well as in establishing the effects of structural bilateral shocks between countries (the so‐called global VAR, or GVAR, models). Special attention is paid to the VECM capacities of being used in conjunction with dynamic stochastic general equilibrium models and of jointly specifying the short‐ and long‐run dynamics, thus representing the steady‐state of economic systems (by means of the co‐integration relations) and the short‐run dynamics around it.     

Inferential Implications of Over‐Parametrization: A Case Study in Incomplete Categorical Data

In the context of either Bayesian or classical sensitivity analyses of over‐parametrized models for incomplete categorical data, it is well known that prior‐dependence on posterior inferences of nonidentifiable parameters or that too parsimonious over‐parametrized models may lead to erroneous conclusions. Nevertheless, some authors either pay no attention to which parameters are nonidentifiable or do not appropriately account for possible prior‐dependence. We review the literature on this topic and consider simple examples to emphasize that in both inferential frameworks, the subjective components can influence results in nontrivial ways, irrespectively of the sample size. Specifically, we show that prior distributions commonly regarded as slightly informative or noninformative may actually be too informative for nonidentifiable parameters, and that the choice of over‐parametrized models may drastically impact the results, suggesting that a careful examination of their effects should be considered before drawing conclusions.     

Forecasting with High‐Dimensional Panel VARs

This paper develops methods for estimating and forecasting in Bayesian panel vector autoregressions of large dimensions with time‐varying parameters and stochastic volatility. We exploit a hierarchical prior that takes into account possible pooling restrictions involving both VAR coefficients and the error covariance matrix, and propose a Bayesian dynamic learning procedure that controls for various sources of model uncertainty. We tackle computational concerns by means of a simulation‐free algorithm that relies on analytical approximations to the posterior. We use our methods to forecast inflation rates in the eurozone and show that these forecasts are superior to alternative methods for large vector autoregressions.     

Marginalized Predictive Likelihood Comparisons of Linear Gaussian State‐Space Models with Applications to DSGE,DSGE‐VAR,and VAR Models

The predictive likelihood is useful for ranking models in forecast comparison exercises using Bayesian inference. We discuss how it can be estimated, by means of marzginalization, for any subset of the observables in linear Gaussian state‐space models. We compare macroeconomic density forecasts for the euro area of a DSGE model to those of a DSGE‐VAR, a BVAR and a multivariate random walk over 1999:Q1–2011:Q4. While the BVAR generally provides superior forecasts, its performance deteriorates substantially with the onset of the Great Recession. This is particularly notable for longer‐horizon real GDP forecasts, where the DSGE and DSGE‐VAR models perform better. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of Hypothesis Testing and Bayesian Model Selection

The main goal of both Bayesian model selection and classical hypotheses testing is to make inferences with respect to the state of affairs in a population of interest. The main differences between both approaches are the explicit use of prior information by Bayesians, and the explicit use of null distributions by the classicists. Formalization of prior information in prior distributions is often difficult. In this paper two practical approaches (encompassing priors and training data) to specify prior distributions will be presented. The computation of null distributions is relatively easy. However, as will be illustrated, a straightforward interpretation of the resulting p-values is not always easy. Bayesian model selection can be used to compute posterior probabilities for each of a number of competing models. This provides an alternative for the currently prevalent testing of hypotheses using p-values. Both approaches will be compared and illustrated using case studies. Each case study fits in the framework of the normal linear model, that is, analysis of variance and multiple regression.