Bayesian inference in Vector Autoregressions (VARs) involves manipulating large matrices which appear in the posterior (or conditional posterior) of the VAR coefficients. For large VARs, the computational burden of these manipulations can render empirical work impractical. In response to this, many researchers transform their VARs so as to allow for Bayesian estimation to proceed one equation at a time. This leads to a massive reduction in the computational burden. This transformation involves taking the Cholesky decomposition for the error covariance matrix. However, this strategy implies that posterior inference depends on the order the variables enter the VAR. In this article we develop an alternative transformation, based on the eigendecomposition, which does not lead to order dependence. Beginning with an inverse-Wishart prior on the error covariance matrix, we derive and discuss the properties of the prior it implies on the eigenmatrix and eigenvalues. We then show how an extension of the prior on the eigenmatrix can allow for greater flexibility while maintaining many of the benefits of conjugacy. We leverage this flexibility to extend the prior on the eigenvalues to allow for stochastic volatility. The properties of the eigendecomposition approach are investigated in a macroeconomic forecasting exercise involving VARs with 20 variables.

This paper proposes a vector autoregression augmented with nonlinear factors that are modeled nonparametrically using regression trees. There are four main advantages of our model. First, modeling potential nonlinearities nonparametrically lessens the risk of misspecification. Second, the use of factor methods ensures that departures from linearity are modeled parsimoniously. In particular, they exhibit functional pooling where a small number of nonlinear factors are used to model common nonlinearities across variables. Third, Bayesian computation using MCMC is straightforward even in very high-dimensional models, allowing for efficient, equation-by-equation estimation, thus avoiding computational bottlenecks that arise in popular alternatives such as the time-varying parameter VAR. Fourth, existing methods for identifying structural economic shocks in linear factor models can be adapted for the nonlinear case in a straightforward fashion using our model. Exercises involving artificial and macroeconomic data illustrate the properties of our model and its usefulness for forecasting and structural economic analysis.

Based on agent opinion analysis theory, Bayesian predictive synthesis (BPS) is a framework for combining predictive distributions in the face of model uncertainty. In this article, we generalize existing parametric implementations of BPS by showing how to combine competing probabilistic forecasts using interpretable Bayesian tree‐based machine learning methods. We demonstrate the advantages of our approach—in terms of improved forecast accuracy and interpretability—via two macroeconomic forecasting applications. The first uses density forecasts for GDP growth from the euro area's Survey of Professional Forecasters. The second combines density forecasts of U.S. inflation produced by many simple regression models.

This paper proposes a vector autoregression augmented with nonlinear factors that are modeled nonparametrically using regression trees. There are four main advantages of our model. First, the use of factor methods ensures that departures from linearity are modeled parsimoniously. In particular, they exhibit functional pooling where a small number of nonlinear factors are used to model common nonlinearities across variables. Second, modeling potential nonlinearities nonparametrically lessens the risk of misspecification. Third, Bayesian computation using MCMC is straightforward even in very high-dimensional models, allowing for efficient, equation-by-equation estimation, thus avoiding computational bottlenecks that arise in popular alternatives such as the time-varying parameter VAR. Fourth, existing methods for identifying structural economic shocks in linear factor models can be adapted for the nonlinear case in a straightforward fashion using our model. Exercises involving artificial and macroeconomic data illustrate the properties of our model and its usefulness for forecasting and structural economic analysis.

This paper develops a mixed frequency vector autoregressive (MF-VAR) model to produce nowcasts and historical estimates of monthly real state-level GDP for the 50 U.S. states, plus Washington DC, from 1964 through the present day. The MF-VAR model incorporates state and U.S. data at the monthly, quarterly, and annual frequencies. Temporal and cross-sectional constraints are imposed to ensure that the monthly state-level estimates are consistent with official estimates of quarterly GDP at the U.S. and state-levels. We illustrate the utility of the historical estimates in better understanding state business cycles and cross-state dependencies. We show how the model produces accurate nowcasts of state GDP three months ahead of the BEA's quarterly estimates, after conditioning on the latest estimates of U.S. GDP.

The macroeconomy is a sophisticated dynamic system involving significant uncertainties that complicate modelling. In response, decision makers consider multiple models that provide different predictions and policy recommendations which are then synthesized into a policy decision. In this setting, we introduce and develop Bayesian predictive decision synthesis (BPDS) to formalize monetary policy decision processes. BPDS draws on recent developments in model combination and statistical decision theory that yield new opportunities in combining multiple models, emphasizing the integration of decision goals, expectations and outcomes into the model synthesis process. Our case study concerns central bank policy decisions about target interest rates with a focus on implications for multi-step macroeconomic forecasting.

Stochastic Volatility in Mean Vector Autoregressions (SVMVARs) are popularly used to jointly estimate uncertainty and its economic effects. However, existing studies analyzing macroeconomic and financial uncertainty require the researcher to classify each variable as macroeconomic or financial before estimation, do not consider whether results are sensitive to model size and adopt a specification where results depend on the way the variables are ordered. We overcome these limitations, developing a novel Markov Chain Monte Carlo algorithm for large, order-invariant SVMVARs with unclassified variables. For each unclassified variable, the algorithm determines the appropriate classification at each point in time. Using a simulation study and large US dataset, we uncover the following. Smaller SVMVARs overstate the effects of uncertainty, failing to reveal that only financial uncertainty has an adverse effect on the economy. When using large order-dependent SVMVARs, however, the uncertainty estimates produced depend on the variable ordering, distorting impulse response analysis. Thus it becomes critical to adopt an order-invariant specification. We also find that many variables change classification with changes often occurring during crisis periods.

Commonly used priors for Vector Autoregressions (VARs) induce shrinkage on the autoregressive coefficients. Introducing shrinkage on the error covariance matrix is sometimes done but, in the vast majority of cases, without considering the network structure of the shocks and by placing the prior on the lower Cholesky factor of the precision matrix. In this paper, we propose a prior on the VAR error precision matrix directly. Our prior, which resembles a standard spike and slab prior, models variable inclusion probabilities through a stochastic block model that clusters shocks into groups. Within groups, the probability of having relations across group members is higher (inducing less sparsity) whereas relations across groups imply a lower probability that members of each group are conditionally related. We show in simulations that our approach recovers the true network structure well. Using a US macroeconomic data set, we illustrate how our approach can be used to cluster shocks together and that this feature leads to improved density forecasts.

The relationship between inflation and predictors such as unemployment is potentially nonlinear with a strength that varies over time, and prediction errors error may be subject to large, asymmetric shocks. Inspired by these concerns, we develop a model for inflation forecasting that is nonparametric both in the conditional mean and in the error using Gaussian and Dirichlet processes, respectively. We discuss how both these features may be important in producing accurate forecasts of inflation. In a forecasting exercise involving CPI inflation, we find that our approach has substantial benefits, both overall and in the left tail, with nonparametric modeling of the conditional mean being of particular importance.

The Bayesian statistical paradigm provides a principled and coherent approach to probabilistic forecasting. Uncertainty about all unknowns that characterize any forecasting problem – model, parameters, latent states – is able to be quantified explicitly and factored into the forecast distribution via the process of integration or averaging. Allied with the elegance of the method, Bayesian forecasting is now underpinned by the burgeoning field of Bayesian computation, which enables Bayesian forecasts to be produced for virtually any problem, no matter how large or complex. The current state of play in Bayesian forecasting in economics and finance is the subject of this review. The aim is to provide the reader with an overview of modern approaches to the field, set in some historical context, with sufficient computational detail given to assist the reader with implementation.

We develop a nonparametric quantile panel regression model. Within each quantile, the quantile function is a combination of linear and nonlinear parts, which we approximate using Bayesian Additive Regression Trees (BART). Cross-sectional information is captured through a conditionally heteroscedastic latent factor. The nonparametric feature enhances flexibility, while the panel feature increases the number of observations in the tails. We develop Bayesian methods for inference and apply several versions of the model to study growth-at-risk dynamics in a panel of 11 advanced economies. Our framework usually improves upon single-country quantile models in recursive growth forecast comparisons.

We develop a Bayesian non-parametric quantile panel regression model. Within each quantile, the response function is a convex combination of a linear model and a non-linear function, which we approximate using Bayesian Additive Regression Trees (BART). Cross-sectional information at the pth quantile is captured through a conditionally heteroscedastic latent factor. The non-parametric feature of our model enhances flexibility, while the panel feature, by exploiting cross-country information, increases the number of observations in the tails. We develop Bayesian Markov chain Monte Carlo (MCMC) methods for estimation and forecasting with our quantile factor BART model (QF-BART), and apply them to study growth at risk dynamics in a panel of 11 advanced economies.

The shocks which hit macroeconomic models such as Vector Autoregressions (VARs) have the potential to be non-Gaussian, exhibiting asymmetries and fat tails. This consideration motivates the VAR developed in this paper which uses a Dirichlet process mixture (DPM) to model the shocks. However, we do not follow the obvious strategy of simply modeling the VAR errors with a DPM since this would lead to computationally infeasible Bayesian inference in larger VARs and potentially a sensitivity to the way the variables are ordered in the VAR. Instead we develop a particular additive error structure inspired by Bayesian nonparametric treatments of random effects in panel data models. We show that this leads to a model which allows for computationally fast and order-invariant inference in large VARs with nonparametric shocks. Our empirical results with nonparametric VARs of various dimensions shows that nonparametric treatment of the VAR errors is particularly useful in periods such as the financial crisis and the pandemic.

Interest in regional economic issues coupled with advances in administrative data is driving the creation of new regional economic data. Many of these data series could be useful for nowcasting regional economic activity, but they suffer from a short (albeit constantly expanding) time series which makes incorporating them into nowcasting models problematic. Regional nowcasting is already challenging because the release delay on regional data tends to be greater than that at the national level, and 'short' data imply a 'ragged edge' at both the beginning and the end of regional data sets, which adds a further complication. In this paper, via an application to the UK, we investigate various ways of including a wide range of short data into a regional mixed-frequency vector autoregression (MF-VAR) model. These short data include hitherto unexploited regional value-added tax turnover data. We address the problem of the two ragged edges by estimating regional factors using different missing data algorithms that we then incorporate into our MF-VAR model. We find that nowcasts of regional output growth are generally improved when we condition them on the factors, but only when the regional nowcasts are produced before the national (UK-wide) output growth data are published.

Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a Cholesky decomposition for the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this Cholesky decomposition. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We develop a Markov Chain Monte Carlo algorithm which allows for Bayesian estimation and prediction. In exercises involving artificial and real macroeconomic data, we demonstrate that the choice of variable ordering can have non-negligible effects on empirical results. In a macroeconomic forecasting exercise involving VARs with 20 variables we find that our order-invariant approach leads to the best forecasts and that some choices of variable ordering can lead to poor forecasts using a conventional, non-order invariant, approach.

Estimating Bayesian Vector Autoregressions (VARs) involving the Cholesky decomposition is sensitive to the ordering of variables. We treat the ordering as unknown, develop a prior over variable orderings and Markov Chain Monte Carlo (MCMC) methods for posterior sampling over orderings.

Using a large international survey of professional forecasters, we construct measures of economic uncertainty surrounding output growth, inflation, the interest rate, exchange rate and current account. We then analyze uncertainty spillovers across major advanced and emerging economies using large multi-country Bayesian Panel VARs. We consider how our results change if our uncertainty measures reflect: disagreement among forecast-ers (idiosyncratic uncertainty); the variance of their mean forecast errors (common uncer-tainty); or both types of uncertainty. We show that the US is an important but not dominant source of uncertainty, affecting other economies through interest rate and exchange rate uncertainty. This reflects the major role played by US monetary policy and the dollar in the global financial system. Crucially, though, the Eurozone followed by the UK and China are also important sources of uncertainty. We also find that, on average, for-eign interest rate and exchange rate uncertainty are more important than foreign output growth uncertainty. While spillovers in idiosyncratic uncertainty are more frequently observed, failing to account for common uncertainty can lead us to overestimate the role played by smaller economies. (c) 2022 Elsevier Ltd. All rights reserved.