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Abstract. This chapter describes Bayesian methods for macroeconomic forecasting using VARs. It covers various priors which have been proposed to achieve the shrinkage and parsimony that is necessary when working with large VARs in different ways. After discussing different conjugate priors and their properties, various non-conjugate priors are proposed including variable selection and global-local shrinkage priors which automatically achieve shrinkage with minimal subjective prior input. A density forecasting exercise using US macroeconomic data illustrates how these methods are implemented in practice as well as their usefulness for forecasting.

Book chapter.

Draft chapter.

R code package.

Abstract. In this paper we estimate a Bayesian vector autoregressive model with factor stochastic volatility in the error term to assess the effects of an uncertainty shock in the Euro area. This allows us to treat macroeconomic uncertainty as a latent quantity during estimation. Only a limited number of contributions to the literature estimate uncertainty and its macroeconomic consequences jointly, and most are based on single country models. We analyze the special case of a shock restricted to the Euro area, where member states are highly related by construction. We find significant results of a decrease in real activity for all countries over a period of roughly a year following an uncertainty shock. Moreover, equity prices, short-term interest rates and exports tend to decline, while unemployment levels increase. Dynamic responses across countries differ slightly in magnitude and duration, with Ireland, Slovakia and Greece exhibiting different reactions for some macroeconomic fundamentals.

Abstract. Empirical evidence has shown that a prerequisite for generating reliable macroeconomic forecasts is either the inclusion of a large information set or modeling time variation in the models’ parameters and volatilities. In this paper we examine these claims in a comparative manner, forecasting GDP growth for six CESEE economies. We use Bayesian techniques and evaluate the models based on both the accuracy of their point forecasts as well as the degree of uncertainty surrounding these predictions. Our results indicate that forecasts from a fully-fledged time-varying parameter model tend to outperform those from its constant parameter competitors. Adding more information, e.g. from other countries, by contrast, does not improve forecast performance significantly for most of the countries under study. Last, we analyze whether it pays to forecast GDP growth indirectly by summing up forecasts of GDP components. This approach yields competitive forecasts, yet it preserves an economic interpretation of the underlying drivers for the economic growth forecasts, which is of crucial importance from a practitioner’s view.

Summary. The large structural decrease in real interest rates presents challenges for policy makers, for instance hindering the ability of central banks to lower policy rates during recessions. We put this decrease in historical perspective for Switzerland by constructing quarterly data going back to 1852 using novel archival sources and statistical methods. We extract long-term trends for interest rates, inflation, and exchange rate growth, and split the nominal interest rate trend into various components. Using an econometric analysis, we relate the real interest rate to monetary regimes, changes in the pension system, international developments, as well as demographic changes. Nominal interest rates have reached historical lows since the Global Financial Crisis. The persistent decline in inflation explains only part of this pattern. The real interest rate, which removes changes in inflation developments, went through an up-down cycle since the 1970's, albeit to a lesser extent than in other countries. During the 19th century Swiss interest rates were above foreign ones, as Switzerland was still an emerging economy at the time. An increase in trend inflation after Switzerland left the Gold Exchange Standard in 1936 pushed nominal interest rates higher. Nevertheless, Switzerland became an "interest rate island" after 1945, as Swiss real interest rates included a negative premium compared to other countries. While this specificity became less important since the 1980's, Swiss nominal interest rates still declined thanks to lower trend inflation and lower foreign interest rates. The decrease was accentuated in 2015, when the term spread between short and long-term rates vanished after the Swiss National Bank abandoned an exchange rate floor policy and the European Central Bank started a large-scale asset purchase program. An econometric analysis shows that the Swiss real interest rate reflects global drivers and demographic factors. Specifically, a higher share of the elderly population lowers the real interest rate, with an opposite effect for the share of the young population. Furthermore, over the last 20 years, the interest rate island vanished, pushing up Swiss rates relative to other countries. While this recent increase can appear surprising, it is important to bear in mind that it captures an effect relative to the rest of the world, where central banks have moved towards monetary policy regimes more conducive to price stability. This may have contributed to a convergence of international to Swiss interest rate trends. While projections are delicate, as long-term interest rate trends can change rapidly, the low inflation trend observed since the Global Financial Crisis, low global real interest rates, and the shift of the age structure of the Swiss population towards a higher share of older people are likely to keep Swiss nominal and real interest rates low. The Covid-19 pandemic may further reinforce this pattern (Jordà et al. 2020). Low interest rates will thus likely remain a challenge for policy makers.

Abstract. The secular increase in globalization led to a substantial increase in the interconnectedness of global financial markets. This has important implications for the conduct of monetary policy, as central bank policies may diverge across countries, potentially affecting key transmission channels of domestic policy actions. In this paper, we use a non-linear multivariate time series model to shed light on how the US monetary policy stance affects the conduct of monetary policy in the euro area. We assume that the dynamic coefficients implicitly depend on a measure of the Federal Reserve's policy stance through a smooth transition function. This assumption allows us to examine how the dynamic responses of financial market quantities such as government bond yields and inflation swaps to euro area monetary policy shocks change with the US policy stance. Scenario-specific impulse responses show that the transmission of euro area monetary policy through financial markets does indeed depend on the prevailing monetary policy regime of the Federal Reserve and has significant effects on a variety of euro area variables.

Abstract. In this paper we aim to improve existing empirical exchange rate models by accounting for uncertainty with respect to the underlying structural representation. Within a flexible Bayesian framework, our modeling approach assumes that different regimes are characterized by commonly used structural exchange rate models, with transitions across regimes being driven by a Markov process. We assume a time-varying transition probability matrix with transition probabilities depending on a measure of the monetary policy stance of the central bank at home and in the USA. We apply this model to a set of eight exchange rates against the US dollar. In a forecasting exercise, we show that model evidence varies over time, and a model approach that takes this empirical evidence seriously yields more accurate density forecasts for most currency pairs considered.

Abstract. This paper proposes a hierarchical modeling approach to perform stochastic model specification in Markov switching vector error correction models. We assume that a common distribution gives rise to the regime-specific regression coefficients. The mean as well as the variances of this distribution are treated as fully stochastic and suitable shrinkage priors are used. These shrinkage priors enable to assess which coefficients differ across regimes in a flexible manner. In the case of similar coefficients, our model pushes the respective regions of the parameter space towards the common distribution. This allows for selecting a parsimonious model while still maintaining sufficient flexibility to control for sudden shifts in the parameters, if necessary. We apply our modeling approach to real-time Euro area data and assume transition probabilities between expansionary and recessionary regimes to be driven by the cointegration errors. The results suggest that the regime allocation is governed by a subset of short-run adjustment coefficients and regime-specific variance-covariance matrices. These findings are complemented by an out-of-sample forecast exercise, illustrating the advantages of the model for predicting Euro area inflation in real time.

Abstract. In line with recent policy discussions on the use of macroprudential policies (MPPs) to respond to cross-border risks arising from capital flows, this paper tries to quantify which impact MPPs had on capital flows in Central, Eastern and Southeastern Europe (CESEE). This region experienced a substantial boom-bust cycle in capital flows amid the global financial crisis and policymakers had been quite active in adopting MPPs already before that crisis. To study the dynamic responses of capital flows to a tightening in the macroprudential environment, we propose a novel regime-switching factor-augmented vector autoregressive (FAVAR) model and include an intensity-adjusted macroprudential policy index to identify MPP shocks. Our results suggest that tighter MPPs translate into negative dynamic reactions of domestic private sector credit growth and gross capital inflow volumes in a majority of the countries analyzed. Level and volatility responses of capital inflows are often correlated positively, suggesting that if MPPs were successful in reducing capital inflows, they would also contribute to lower capital flow volatility. We also provide evidence that the effects of MPP tightening are in most cases stronger in an environment characterized by low interest rates, suggesting that MPPs would be more effective if conventional monetary policy were facing constraints.

Abstract. Conjugate priors allow for fast inference in large dimensional vector autoregressive (VAR) models. But at the same time, they introduce the restriction that each equation features the same set of explanatory variables. This paper proposes a straightforward means of postprocessing posterior estimates of a conjugate Bayesian VAR to effectively perform equation-specific covariate selection. Compared with existing techniques using shrinkage alone, our approach combines shrinkage and sparsity in both the VAR coefficients and the error variance–covariance matrices, greatly reducing estimation uncertainty in large dimensions while maintaining computational tractability. We illustrate our approach by means of two applications. The first application uses synthetic data to investigate the properties of the model across different data-generating processes, and the second application analyzes the predictive gains from sparsification in a forecasting exercise for U.S. data.

Abstract. Time-varying parameter (TVP) models often assume that the TVPs evolve according to a random walk. This assumption, however, might be questionable since it implies that coefficients change smoothly and in an unbounded manner. This assumption is relaxed by proposing a flexible law of motion for the TVPs in large-scale vector autoregressions (VARs). Instead of imposing a restrictive random walk evolution of the latent states, hierarchical mixture priors on the coefficients in the state equation are carefully designed. These priors effectively allow for discriminating between periods in which coefficients evolve according to a random walk and times where the TVPs are better characterized by a stationary stochastic process. Moreover, this approach is capable of introducing dynamic sparsity by pushing small parameter changes towards zero if necessary. The merits of the model are illustrated by means of two applications. Using synthetic data these flexible modeling techniques yield precise parameter estimates. When applied to US data, the model reveals interesting patterns of low-frequency dynamics in coefficients and forecasts well relative to a wide range of competing models.

Abstract. We extend the econometric literature on the role of production networks in the propagation of monetary policy shocks along two dimensions. First, we allow for time-varying industry-specific responses, reflecting non-linearities and heterogeneity in direct transmission channels. Second, we allow for time-varying network structures and dependence. This captures both variation in the structure of the production network and differences in cross-industry demand elasticities. Spillover effects among industries appear to be important in periods of elevated economic and financial uncertainty, often coinciding with tight credit market conditions and financial stress. Cross-sectional differentials can be explained by how close industries are to end-consumers.

Abstract. In this paper, we investigate the effectiveness of conventional and unconventional monetary policy measures by the European Central Bank (ECB) conditional on the prevailing level of uncertainty. To obtain exogenous variation in central bank policy, we rely on high-frequency surprises in financial market data for the euro area (EA) around policy announcement dates. We trace the dynamic effects of shocks to the short-term policy rate, forward guidance and quantitative easing on several key macroeconomic and financial quantities alongside survey-based measures of expectations. For this purpose, we propose a Bayesian smooth-transition vector autoregression (ST-VAR), using a measure of economic policy uncertainty as signal variable. Our results suggest that transmission channels are impaired when uncertainty is elevated. While conventional monetary policy and forward guidance can be less effective during such periods, quantitative easing measures seem to work comparatively well in uncertain times.

Abstract. In this article, we write the time-varying parameter (TVP) regression model involving K explanatory variables and T observations as a constant coefficient regression model with KT explanatory variables. In contrast with much of the existing literature which assumes coefficients to evolve according to a random walk, a hierarchical mixture model on the TVPs is introduced. The resulting model closely mimics a random coefficients specification which groups the TVPs into several regimes. These flexible mixtures allow for TVPs that feature a small, moderate or large number of structural breaks. We develop computationally efficient Bayesian econometric methods based on the singular value decomposition of the KT regressors. In artificial data, we find our methods to be accurate and much faster than standard approaches in terms of computation time. In an empirical exercise involving inflation forecasting using a large number of predictors, we find our models to forecast better than alternative approaches and document different patterns of parameter change than are found with approaches which assume random walk evolution of parameters.

Abstract. We provide a comprehensive assessment of the predictive power of combinations of dynamic stochastic general equilibrium (DSGE) models for GDP growth, inflation, and the interest rate in the euro area. We employ a battery of static and dynamic pooling weights based on Bayesian model averaging principles, prediction pools, and dynamic factor representations, and entertain six different DSGE specifications and five prediction weighting schemes. Our results indicate that exploiting mixtures of DSGE models produces competitive forecasts compared to individual specifications for both point and density forecasts over the last three decades. Although these combinations do not tend to systematically achieve superior forecast performance, we find improvements for particular periods of time and variables when using prediction pooling, dynamic model averaging, and combinations of forecasts based on Bayesian predictive synthesis.

Abstract. In this paper, we assess whether using non-linear dimension reduction techniques pays off for forecasting inflation in real-time. Several recent methods from the machine learning literature are adopted to map a large dimensional dataset into a lower-dimensional set of latent factors. We model the relationship between inflation and the latent factors using constant and time-varying parameter (TVP) regressions with shrinkage priors. Our models are then used to forecast monthly US inflation in real-time. The results suggest that sophisticated dimension reduction methods yield inflation forecasts that are highly competitive with linear approaches based on principal components. Among the techniques considered, the Autoencoder and squared principal components yield factors that have high predictive power for one-month- and one-quarter-ahead inflation. Zooming into model performance over time reveals that controlling for non-linear relations in the data is of particular importance during recessionary episodes of the business cycle or the current COVID-19 pandemic.

Abstract. US yield curve dynamics are subject to time-variation, but there is ambiguity about its precise form. This paper develops a vector autoregressive (VAR) model with time-varying parameters and stochastic volatility, which treats the nature of parameter dynamics as unknown. Coefficients can evolve according to a random walk, a Markov switching process, observed predictors, or depend on a mixture of these. To decide which form is supported by the data and to carry out model selection, we adopt Bayesian shrinkage priors. Our framework is applied to model the US yield curve. We show that the model forecasts well, and focus on selected in-sample features to analyze determinants of structural breaks in US yield curve dynamics.

Abstract. 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.

Abstract. We develop a non-parametric multivariate time series model that remains agnostic on the precise relationship between a (possibly) large set of macroeconomic time series and their lagged values. The main building block of our model is a Gaussian process prior on the functional relationship that determines the conditional mean of the model, hence the name of Gaussian process vector autoregression (GP-VAR). A flexible stochastic volatility specification is used to provide additional flexibility and control for heteroskedasticity. Markov chain Monte Carlo (MCMC) estimation is carried out through an efficient and scalable algorithm which can handle large models. The GP-VAR is illustrated by means of simulated data and in a forecasting exercise with US data. Moreover, we use the GP-VAR to analyze the effects of macroeconomic uncertainty, with a particular emphasis on time variation and asymmetries in the transmission mechanisms.

Abstract. In this paper we propose a time-varying parameter (TVP) vector error correction model (VECM) with heteroskedastic disturbances. We propose tools to carry out dynamic model specification in an automatic fashion. This involves using global-local priors, and postprocessing the parameters to achieve truly sparse solutions. Depending on the respective set of coefficients, we achieve this via minimizing auxiliary loss functions. Our two-step approach limits overfitting and reduces parameter estimation uncertainty. We apply this framework to modeling European electricity prices. When considering daily electricity prices for different markets jointly, our model highlights the importance of explicitly addressing cointegration and nonlinearities. In a forecasting exercise focusing on hourly prices for Germany, our approach yields competitive metrics of predictive accuracy.

Abstract. We propose a method to learn the nonlinear impulse responses to structural shocks using neural networks, and apply it to uncover the effects of US financial shocks. The results reveal substantial asymmetries with respect to the sign of the shock. Adverse financial shocks have powerful effects on the US economy, while benign shocks trigger much smaller reactions. Instead, with respect to the size of the shocks, we find no discernible asymmetries.

Abstract. Macroeconomic data is characterized by a limited number of observations (small T), many time series (big K) but also by featuring temporal dependence. Neural networks, by contrast, are designed for datasets with millions of observations and covariates. In this paper, we develop Bayesian neural networks (BNNs) that are well-suited for handling datasets commonly used for macroeconomic analysis in policy institutions. Our approach avoids extensive specification searches through a novel mixture specification for the activation function that appropriately selects the form of nonlinearities. Shrinkage priors are used to prune the network and force irrelevant neurons to zero. To cope with heteroskedasticity, the BNN is augmented with a stochastic volatility model for the error term. We illustrate how the model can be used in a policy institution through simulations and by showing that BNNs produce more accurate point and density forecasts compared to other machine learning methods.

Abstract. 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 paper, 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 US inflation produced by many simple regression models.

Abstract. Timely and accurate information about regional economic conditions can be essential for planning, implementing and evaluating locally targeted economic policies. However, European regional accounts for output are published at annual frequency with a two-year delay. To obtain robust and more timely measures in a computationally efficient manner, we propose a mixed-frequency dynamic factor model that accounts for national information to produce high-frequency estimates and reliable nowcasts of regional gross value added (GVA) in more than 150 regions across 13 European countries.

Abstract. We study domestic and international drivers of long-term interest rates using newly compiled financial market data for Switzerland starting in 1852. We use a time-varying parameter vector autoregressive model to estimate long-term trends in nominal interest rates, exchange rate growth, and inflation. We then decompose the Swiss long-term interest rate trend into various drivers using an interest rate accounting framework. The decline in long-term interest rates since 1970 is mainly driven by a decline in the level of inflation. Comparing Switzerland with the rest of the world, we show that while Swiss real interest rates were higher during the 19th century, the pattern reversed after WorldWar 2 with Swiss nominal and real rates becoming lower than foreign ones. However, this Swiss "low interest rate island" has disappeared in recent years. We document a connection between inflation risk and the Swiss term spread, as well between relative inflation risk and the difference between Swiss and foreign real interest rates.

Abstract. In light of widespread evidence of parameter instability in macroeconomic models, many time-varying parameter (TVP) models have been proposed. This paper proposes a nonparametric TVP-VAR model using Bayesian Additive Regression Trees (BART). The novelty of this model arises from the law of motion driving the parameters being treated nonparametrically. This leads to great flexibility in the nature and extent of parameter change, both in the conditional mean and in the conditional variance. In contrast to other nonparametric and machine learning methods that are black box, inference using our model is straightforward because, in treating the parameters rather than the variables nonparametrically, the model remains conditionally linear in the mean. Parsimony is achieved through adopting nonparametric factor structures and use of shrinkage priors. In an application to US macroeconomic data, we illustrate the use of our model in tracking both the evolving nature of the Phillips curve and how the effects of business cycle shocks on inflationary measures vary nonlinearly with movements in uncertainty.

Abstract. We propose and discuss Bayesian machine learning methods for mixed data sampling (MIDAS) regressions. This involves handling frequency mismatches with restricted and unrestricted MIDAS variants and specifying functional relationships between many predictors and the dependent variable. We use Gaussian processes (GP) and Bayesian additive regression trees (BART) as flexible extensions to linear penalized estimation. In a nowcasting and forecasting exercise we focus on quarterly US output growth and inflation in the GDP deflator. The new models leverage macroeconomic Big Data in a computationally efficient way and offer gains in predictive accuracy along several dimensions.

Abstract. We propose a Bayesian Asset Pricing framework that uses machine learning to capture nonlinear interactions between firm exposures to climate change risks and a number of possible effect modifiers. Climate change risks are measured by two indicators that are obtained through textual analysis. Considering a portfolio of 430 US stocks, we find that news about climate change impact stock returns nonlinearly and asymmetrically. Our findings indicate that sectors with high pollution levels are most affected, with the highest polluting firms within these sectors bearing the brunt of the impact. In addition, we find that the insurance and transportation sectors are most affected by natural disaster news.

Abstract. We propose a dynamic factor model (DFM) where the latent factors are linked to observed variables with unknown and potentially nonlinear functions. The key novelty and source of flexibility of our approach is a nonparametric observation equation, specified via Gaussian Process (GP) priors for each series. Factor dynamics are modeled with a standard vector autoregression (VAR), which facilitates computation and interpretation. We discuss a computationally efficient estimation algorithm and consider two empirical applications. First, we forecast key series from the FRED-QD dataset and show that the model yields improvements in predictive accuracy relative to linear benchmarks. Second, we extract driving factors of global inflation dynamics with the GP-DFM, which allows for capturing international asymmetries.