Volume 28, Issue 2 - Special Issue in Honor of Herman van Dijk; Guest Editors: Gary Koop, Dimitris Korobilis and Francesco Ravazzolo

This essay is about Bayesian econometrics with a purpose . Specifically, six societal challenges and research opportunities that confront twenty first century Bayesian econometricians are discussed using an important feature of modern Bayesian econometrics: conditional probabilities of a wide range of economic events of interest can be evaluated by using simulation-based Bayesian inference. The enormous advances in hardware and software have made this Bayesian computational approach a very attractive vehicle of research in many subfields in economics where novel data patterns and substantial model complexity are predominant. In this essay the following challenges and opportunities are briefly discussed, including the scientific results obtained in the twentieth century leading up to these challenges: Posterior and predictive analysis of everything: connecting micro-economic causality with macro-economic issues; the need for speed: model complexity and the golden age of algorithms; learning about models, forecasts and policies including their uncertainty; temporal distributional change due to polarisation, imbalances and shocks; climate change and the macroeconomy; finally and most importantly, widespread, accessible, advanced high-level training.

The basic Markov-switching model has been extended in various ways ever since the seminal work of Hamilton (1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57: 357–84). However, the estimation of Markov-switching models in the literature has relied upon parametric assumptions on the distribution of the error term. In this paper, we present a Bayesian approach for estimating Markov-switching models with unknown and potentially non-normal error distributions. We approximate the unknown distribution of the error term by the Dirichlet process mixture of normals, in which the number of mixtures is treated as a parameter to estimate. In doing so, we pay special attention to the identification of the model. We then apply the proposed model and MCMC procedure to the growth of the postwar U.S. industrial production index. Our model can effectively control for irregular components that are not related to business conditions. This leads to sharp and accurate inferences on recession probabilities.

Time-varying parameter (TVP) regression models can involve a huge number of coefficients. Careful prior elicitation is required to yield sensible posterior and predictive inferences. In addition, the computational demands of Markov Chain Monte Carlo (MCMC) methods mean their use is limited to the case where the number of predictors is not too large. In light of these two concerns, this paper proposes a new dynamic shrinkage prior which reflects the empirical regularity that TVPs are typically sparse (i.e. time variation may occur only episodically and only for some of the coefficients). A scalable MCMC algorithm is developed which is capable of handling very high dimensional TVP regressions or TVP Vector Autoregressions. In an exercise using artificial data we demonstrate the accuracy and computational efficiency of our methods. In an application involving the term structure of interest rates in the eurozone, we find our dynamic shrinkage prior to effectively pick out small amounts of parameter change and our methods to forecast well.

The issue of modelling observations generated in matrix form over time is key in economics, finance and many domains of application. While it is common to model vectors of observations through standard vector time series analysis, original matrix-valued data often reflect different types of structures of time series observations which can be further exploited to model interdependencies. In this paper, we propose a novel matrix autoregressive model in a bilinear form which, while leading to a substantial dimensionality reduction and enhanced interpretability: (a) allows responses and potential covariates of interest to have different dimensions; (b) provides a suitable estimation procedure for matrix autoregression with lag structure; (c) facilitates the introduction of Bayesian estimators. We propose maximum likelihood and Bayesian estimation with Independent-Normal prior formulation, and study the theoretical properties of the estimators through simulated and real examples.

Modern macroeconometrics often relies on time series models for which it is time-consuming to evaluate the likelihood function. We demonstrate how Bayesian computations for such models can be drastically accelerated by reweighting and mutating posterior draws from an approximating model that allows for fast likelihood evaluations, into posterior draws from the model of interest, using a sequential Monte Carlo (SMC) algorithm. We apply the technique to the estimation of a vector autoregression with stochastic volatility and two nonlinear dynamic stochastic general equilibrium models. The runtime reductions we obtain range from 27 % to 88 %.

This paper investigates the determinants of the European iTraxx corporate CDS index considering a large set of explanatory variables within a Markov switching model framework. The influence of financial and economic variables on CDS spreads are compared using linear, two, three, and four-regime models in a sample post-subprime financial crisis up to the COVID-19 pandemic. Results indicate that four regimes are necessary to model the CDS spreads. The fourth regime was activated during the COVID-19 pandemic and in high volatility periods. Further, the effect of the covariates differs significantly across regimes. Brent and term structure factors became relevant after the outbreak of the COVID-19 pandemic.

Bayesian Predictive Synthesis is a flexible method of combining density predictions. The flexibility comes from the ability to choose an arbitrary synthesis function to combine predictions. I study choice of synthesis function when combining large numbers of predictions – a common occurrence in macroeconomics. Estimating combination weights with many predictions is difficult, so I consider shrinkage priors and factor modelling techniques to address this problem. These techniques provide an interesting contrast between the sparse weights implied by shrinkage priors and dense weights of factor modelling techniques. I find that the sparse weights of shrinkage priors perform well across exercises.

The paper examines the question of non-anonymous Growth Incidence Curves (na-GIC) from a Bayesian inferential point of view. Building on the notion of conditional quantiles of Barnett (1976. “The Ordering of Multivariate Data.” Journal of the Royal Statistical Society: Series A 139: 318–55), we show that removing the anonymity axiom leads to a complex and shaky curve that has to be smoothed, using a non-parametric approach. We opted for a Bayesian approach using Bernstein polynomials which provides confidence intervals, tests and a simple way to compare two na-GICs. The methodology is applied to examine wage dynamics in a US university with a particular attention devoted to unbundling and anti-discrimination policies. Our findings are the detection of wage scale compression for higher quantiles for all academics and an apparent pro-female wage increase compared to males. But this pro-female policy works only for academics and not for the para-academics categories created by the unbundling policy.

This article considers a stable vector autoregressive (VAR) model and investigates return predictability in a Bayesian context. The bivariate VAR system comprises asset returns and a further prediction variable, such as the dividend-price ratio, and allows pinning down the question of return predictability to the value of one particular model parameter. We develop a new shrinkage type prior for this parameter and compare our Bayesian approach to ordinary least squares estimation and to the reduced-bias estimator proposed in Amihud and Hurvich (2004. “Predictive Regressions: A Reduced-Bias Estimation Method.” Journal of Financial and Quantitative Analysis 39: 813–41). A simulation study shows that the Bayesian approach dominates the reduced-bias estimator in terms of observed size (false positive) and power (false negative). We apply our methodology to a system comprising annual CRSP value-weighted returns running, respectively, from 1926 to 2004 and from 1953 to 2021, and the logarithmic dividend-price ratio. For the first sample, the Bayesian approach supports the hypothesis of no return predictability, while for the second data set weak evidence for predictability is observed. Then, instead of the dividend-price ratio, some prediction variables proposed in Welch and Goyal (2008. “A Comprehensive Look at the Empirical Performance of Equity Premium Prediction.” Review of Financial Studies 21: 1455–508) are used. Also with these prediction variables, only weak evidence for return predictability is supported by Bayesian testing. These results are corroborated with an out-of-sample forecasting analysis.

Periods of financial turmoil are not only characterized by higher correlation across assets but also by modifications in their overall clustering structure. In this work, we develop a dynamic Latent-Space mixture model for capturing changes in the clustering structure of financial assets at a fine scale. Through this model, we are able to project stocks onto a lower dimensional manifold and detect the presence of clusters. The infinite-mixture assumption ensures tractability in inference and accommodates cases in which the number of clusters is large. The Bayesian framework we rely on accounts for uncertainty in the parameters’ space and allows for the inclusion of prior knowledge. After having tested our model’s effectiveness and inference on a suitable synthetic dataset, we apply the model to the cross-correlation series of two reference stock indices. Our model correctly captures the presence of time-varying asset clustering. Moreover, we notice how assets’ latent coordinates may be related to relevant financial factors such as market capitalization and volatility. Finally, we find further evidence that the number of clusters seems to soar in periods of financial distress.

Increases in the use of Bayesian inference in applied analysis, the complexity of estimated models, and the popularity of efficient Markov chain Monte Carlo (MCMC) inference under conjugate priors have led to more scrutiny regarding the specification of the parameters in prior distributions. Impact of prior parameter assumptions on posterior statistics is commonly investigated in terms of local or pointwise assessments, in the form of derivatives or more often multiple evaluations under a set of alternative prior parameter specifications. This paper expands upon these localized strategies and introduces a new approach based on the graph of posterior statistics over prior parameter regions (sensitivity manifolds) that offers additional measures and graphical assessments of prior parameter dependence. Estimation is based on multiple point evaluations with Gaussian processes, with efficient selection of evaluation points via active learning, and is further complemented with derivative information. The application introduces a strategy to assess prior parameter dependence in a multivariate demand model with a high dimensional prior parameter space, where complex prior-posterior dependence arises from model parameter constraints. The new measures uncover a considerable prior dependence beyond parameters suggested by theory, and reveal novel interactions between the prior parameters and the elasticities.

We develop a methodology to estimate impulse response functions via Bayesian techniques with the goal of providing a bridge between a linear vector autoregressive specification and a high-order polynomial local projection, namely flexible local projection. We label this methodology Bayesian Flexible Local Projection (BFLP). We assess the properties of BFLP in a Monte Carlo framework considering both linear and non-linear models as data generating processes. We also empirically illustrate how BFLP can be used with standard identification strategies. In particular, we show how to use external instruments to identify the effects of the monetary policy shock in the United States. Furthermore, exploiting the time-varying nature of the impulse response functions based on BFLP, we assess the zero lower bound irrelevance hypothesis and find no strong evidence that monetary policy was less effective in influencing output and inflation during the recent ZLB period.