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.

Vector autoregressions (VARs) are popular for forecasting, but ill-suited to handle occasionally binding constraints, like the effective lower bound on nominal interest rates. We examine reduced-form "shadow rate VARs" that model interest rates as censored observations of a latent shadow rate process and develop an efficient Bayesian estimation algorithm that accommodates large models. When compared to a standard VAR, our better-performing shadow rate VARs generate superior predictions for interest rates and broadly similar predictions for macroeconomic variables. We obtain this result for shadow rate VARs in which the federal funds rate is the only interest rate and in models including additional interest rates. Our shadow rate VARs also deliver notable gains in forecast accuracy relative to a VAR that omits shorter-term interest rate data in order to avoid modeling the lower bound.

This paper examines the forecasting efficacy and implications of the recently popular breakdown of core inflation into three components: goods excluding food and energy, services excluding energy and housing, and housing. A comprehensive historical evaluation of the accuracy of point and density forecasts from a range of models and approaches shows that a BVAR with stochastic volatility in aggregate core inflation, its three components, and wage growth is an effective tool for forecasting inflation's components as well as aggregate core inflation. Looking ahead, the model's baseline projection puts core inflation at 2.6 percent in 2026, well below its 2023 level but still elevated relative to the Federal Reserve's 2 percent objective. The probability that core inflation will return to 2 percent or less is much higher when conditioning on goods or non-housing services inflation slowing to pre-pandemic levels than when conditioning on these components remaining above the same thresholds. Scenario analysis indicates that slower wage growth will likely be associated with reduced inflation in all three components, especially goods and non-housing services, helping to return core inflation to near the 2 percent target by 2026.

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.

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.

A rapidly growing body of research has examined tail risks in macroeconomic outcomes. Most of this work has focused on the risks of significant declines in GDP, and it has relied on quantile regression methods to estimate tail risks. Although much of this work discusses asymmetries in conditional predictive distributions, the analysis often focuses on evidence of downside risk varying more than upside risk. We note that this pattern in risk estimates over time could obtain with conditional distributions that are symmetric but subject to simultaneous shifts in conditional means (down) and variances (up). Building on that insight, we examine the ability of Bayesian VARs with stochastic volatility to capture tail risks in macroeconomic forecast distributions and outcomes. We consider both a conventional stochastic volatility specification and a specification with a common factor in volatility that enters the VAR’s conditional mean. Even though the one-step-ahead conditional predictive distributions from the conventional stochastic volatility specification are symmetric, the model estimates yield more time variation in downside risk as compared to upside risk. Results from the model that includes a volatility factor in the conditional mean and thereby allows for asymmetries in conditional distributions are very similar. Our paper also extends the recent literature by formally evaluating the accuracy of tail risk forecasts and assessing the performance of Bayesian quantile regression, as well as our Bayesian VARs, in this context. Overall, the BVAR models perform comparably to quantile regression for estimating and forecasting tail risks, complementing BVARs’ established performance for forecasting and structural analysis.

Quantile regression has become widely used in empirical macroeconomics, in particular for estimating and forecasting tail risks to macroeconomic indicators. In this paper we examine various choices in the specification of quantile regressions for macro applications, for example, choices related to how and to what extent to include shrinkage, and whether to apply shrinkage in a classical or Bayesian framework. We focus on forecasting accuracy, using for evaluation both quantile scores and quantile-weighted continuous ranked probability scores at a range of quantiles spanning from the left to right tail. We find that shrinkage is generally helpful to tail forecast accuracy, with gains that are particularly large for GDP applications featuring large sets of predictors and unemployment and inflation applications, and with gains that increase with the forecast horizon.

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 COVID-19 pandemic has led to enormous data movements that strongly affect parameters and forecasts from standard Bayesian vector autoregressions (BVARs). To address these issues, we propose BVAR models with outlier-augmented stochastic volatility (SV) that combine transitory and persistent changes in volatility. The resulting density forecasts are much less sensitive to outliers in the data than standard BVARs. Predictive Bayes factors indicate that our outlier-augmented SV model provides the best fit for the pandemic period, as well as for earlier subsamples of high volatility. In historical forecasting, outlier-augmented SV schemes fare at least as well as a conventional SV model.

Since early 2021, inflation has consistently exceeded the Federal Reserve’s target of 2 percent. Using a combination of data, economic theory, and narrative information around historical events, we empirically assess what has caused persistently elevated inflation. Our estimates suggest that both aggregate demand and supply factors, including supply chain disruptions, have contributed significantly to high inflation.

We develop multivariate time-series models using Bayesian additive regression trees that posit nonlinearities among macroeconomic variables, their lags, and possibly their lagged errors. The error variances can be stable, feature stochastic volatility, or follow a nonparametric specification. We evaluate density and tail forecast performance for a set of U.S. macroeconomic and financial indicators. Our results suggest that the proposed models improve forecast accuracy both overall and in the tails. Another finding is that when allowing for nonlinearities in the conditional mean, heteroskedasticity becomes less important. A scenario analysis reveals nonlinear relations between predictive distributions and financial conditions.

VARs are a popular tool for forecasting and structural analysis, but ill-suited to handle occasionally binding constraints, like the effective lower bound on nominal interest rates. We extend the VAR framework by modeling interest rates as censored observations of a latent shadow-rate process, and propose an efficient sampler for Bayesian estimation of such 'shadow-rate VARs.' We analyze specifications where actual and shadow rates serve as explanatory variables and find benefits of including both. In comparison to a standard VAR, shadow-rate VARs generate superior predictions for short- and long-term interest rates, and deliver some gains for macroeconomic variables in US data. Our structural analysis estimates economic responses to shocks in financial conditions, showing strong differences in the reaction of interest rates depending on whether the ELB binds or not. After an adverse shock, our shadow-rate VAR sees a stronger decline of economic activity at the ELB rather than when not.

Abstract Vector autoregressions with stochastic volatility (SV) are widely used in macroeconomic forecasting and structural inference. The SV component of the model conveniently allows for time variation in the variance-covariance matrix of the model’s forecast errors. In turn, that feature of the model generates time variation in predictive densities. The models are most commonly estimated with Bayesian methods, most typically Markov chain Monte Carlo methods, such as Gibbs sampling. Equation-by-equation methods developed since 2018 enable the estimation of models with large variable sets at much lower computational cost than the standard approach of estimating the model as a system of equations. The Bayesian framework also facilitates the accommodation of mixed frequency data, non-Gaussian error distributions, and nonparametric specifications. With advances made in the 21st century, researchers are also addressing some of the framework’s outstanding challenges, particularly the dependence of estimates on the ordering of variables in the model and reliable estimation of the marginal likelihood, which is the fundamental measure of model fit in Bayesian methods.

The original algorithm contained a mistake that meant the conditional distributions used for the VAR’s coefficients were missing a piece of information. We propose a new algorithm that uses the same factorization but includes the missing term. The new, correct algorithm has the same computational complexity as the old, incorrect one (i.e., O(N4)), and therefore it still allows the estimation of large VARs.

Carriero, Clark, and Marcellino (2020, CCM2020) used large BVAR models with a factor structure to stochastic volatility to produce estimates of timevarying international macroeconomic uncertainty and assess uncertainty’s e↵ects on the global economy. The results in CCM2020 were based on an estimation algorithm that has recently been shown to be incorrect by Bognanni (2021) and fixed by Carriero, et al. (2021). In this note we use the algorithm correction of Carriero, et al. (2021) to correct the estimates of CCM2020. Although the correction has some impact on the original results, the changes are small and the key findings of CCM2020 are upheld.

SummaryThis paper focuses on nowcasts of tail risk to GDP growth, with a potentially wide array of monthly and weekly information used to produce nowcasts on a weekly basis. We consider Bayesian mixed frequency regressions with stochastic volatility and Bayesian quantile regressions. Our results show that, within some limits, more information helps the accuracy of nowcasts of tail risk to GDP growth. Accuracy typically improves as time moves forward within a quarter, making additional data available, with monthly data more important to accuracy than weekly data. Accuracy also typically improves with the use of financial indicators in addition to a base set of macroeconomic indicators.

We develop a model that permits the estimation of a term structure of both expectations and forecast uncertainty for application to professional forecasts such as the Survey of Professional Forecasters (SPF). Our approach exactly replicates a given data set of predictions from the SPF (or a similar forecast source) without measurement error. Our model captures fixed horizon and fixed-event forecasts, and can accommodate changes in the maximal forecast horizon available from the SPF. The model casts a decomposition of multi-period forecast errors into a sequence of forecast updates that may be partially unobserved, resulting in a multivariate unobserved components model. In our empirical analysis, we provide quarterly term structures of expectations and uncertainty bands. Our preferred specification features stochastic volatility in forecast updates, which improves forecast performance and yields model estimates of forecast uncertainty that vary over time. We conclude by constructing SPF-based fan charts for calendar-year forecasts like those published by the Federal Reserve.Replication files are available at https://github.com/elmarmertens/ClarkGanicsMertensSPFfancharts.

We propose a new framework for measuring uncertainty and its effects on the economy, based on a large VAR model with errors whose stochastic volatility is driven by two common unobservable factors, representing aggregate macroeconomic and financial uncertainty. The uncertainty measures can also influence the levels of the variables so that, contrary to most existing measures, ours reflect changes in both the conditional mean and volatility of the variables, and their impact on the economy can be assessed within the same framework. Moreover, identification of the uncertainty shocks is simplified with respect to standard VAR-based analysis, in line with the FAVAR approach and with heteroskedasticity-based identification. Finally, the model, which is also applicable in other contexts, is estimated with a new Bayesian algorithm, which is computationally efficient and allows for jointly modeling many variables, while previous VAR models with stochastic volatility could only handle a handful of variables. Empirically, we apply the method to estimate uncertainty and its effects using US data, finding that there is indeed substantial commonality in uncertainty, sizable effects of uncertainty on key macroeconomic and financial variables with responses in line with economic theory.

In this paper we propose a hierarchical shrinkage approach for multi-country VAR models. In implementation, we consider three different scale mixtures of Normals priors — specifically, Horseshoe, NormalGamma, and Normal-Gamma-Gamma priors. We provide new theoretical results for the Normal-Gamma prior. Empirically, we use a quarterly data set for the G7 economies to examine how model specifications and prior choices affect the forecasting performance for GDP growth, inflation, and a short-term interest rate. We find that hierarchical shrinkage, particularly as implemented with the Horseshoe prior, is very useful in forecasting inflation. It also has the best density forecast performance for output growth and the interest rate. Adding foreign information yields benefits, as multi-country models generally improve on the forecast accuracy of single-country models.