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## Bayesian Vector Autoregressions: Estimation

Vector autoregressions (VARs) are linear multivariate time-series models able to capture the joint dynamics of multiple time series. Bayesian inference treats the VAR parameters as random variables, and it provides a framework to estimate “posterior” probability distribution of the location of the model parameters by combining information provided by a sample of observed data and prior information derived from a variety of sources, such as other macro or micro datasets, theoretical models, other macroeconomic phenomena, or introspection. In empirical work in economics and finance, informative prior probability distributions are often adopted. These are intended to summarize stylized representations of the data generating process. For example, “Minnesota” priors, one of the most commonly adopted macroeconomic priors for the VAR coefficients, express the belief that an independent random-walk model for each variable in the system is a reasonable “center” for the beliefs about their time-series behavior. Other commonly adopted priors, the “single-unit-root” and the “sum-of-coefficients” priors are used to enforce beliefs about relations among the VAR coefficients, such as for example the existence of co-integrating relationships among variables, or of independent unit-roots. Priors for macroeconomic variables are often adopted as “conjugate prior distributions”—that is, distributions that yields a posterior distribution in the same family as the prior p.d.f.—in the form of Normal-Inverse-Wishart distributions that are conjugate prior for the likelihood of a VAR with normally distributed disturbances. Conjugate priors allow direct sampling from the posterior distribution and fast estimation. When this is not possible, numerical techniques such as Gibbs and Metropolis-Hastings sampling algorithms are adopted. Bayesian techniques allow for the estimation of an ever-expanding class of sophisticated autoregressive models that includes conventional fixed-parameters VAR models; Large VARs incorporating hundreds of variables; Panel VARs, that permit analyzing the joint dynamics of multiple time series of heterogeneous and interacting units. And VAR models that relax the assumption of fixed coefficients, such as time-varying parameters, threshold, and Markov-switching VARs.

## Bayesian Vector Autoregressions: Applications

Bayesian vector autoregressions (BVARs) are standard multivariate autoregressive models routinely used in empirical macroeconomics and finance for structural analysis, forecasting, and scenario analysis in an ever-growing number of applications. A preeminent field of application of BVARs is forecasting. BVARs with informative priors have often proved to be superior tools compared to standard frequentist/flat-prior VARs. In fact, VARs are highly parametrized autoregressive models, whose number of parameters grows with the square of the number of variables times the number of lags included. Prior information, in the form of prior distributions on the model parameters, helps in forming sharper posterior distributions of parameters, conditional on an observed sample. Hence, BVARs can be effective in reducing parameters uncertainty and improving forecast accuracy compared to standard frequentist/flat-prior VARs. This feature in particular has favored the use of Bayesian techniques to address “big data” problems, in what is arguably one of the most active frontiers in the BVAR literature. Large-information BVARs have in fact proven to be valuable tools to handle empirical analysis in data-rich environments. BVARs are also routinely employed to produce conditional forecasts and scenario analysis. Of particular interest for policy institutions, these applications permit evaluating “counterfactual” time evolution of the variables of interests conditional on a pre-determined path for some other variables, such as the path of interest rates over a certain horizon. The “structural interpretation” of estimated VARs as the data generating process of the observed data requires the adoption of strict “identifying restrictions.” From a Bayesian perspective, such restrictions can be seen as dogmatic prior beliefs about some regions of the parameter space that determine the contemporaneous interactions among variables and for which the data are uninformative. More generally, Bayesian techniques offer a framework for structural analysis through priors that incorporate uncertainty about the identifying assumptions themselves.

## Shocks, Information, and Structural VARs

Structural Vector Autoregressions (SVARs) have become one of the most popular tools to measure the effects of structural economic shocks. Several new techniques to “identify” economic shocks have been proposed in the literature in the last decades. Identification hinges on the implicit assumption that economic shocks are retrievable from the data. In other words, the data contain enough information to correctly estimate the shocks. SVAR models, however, are small-scale models, only a small number of variables can be handled, and this feature can forcefully limit the amount of information that variables can convey. Narrow information sets present problems for identification, but some theoretical results and empirical procedures can test whether such information is sufficient to estimate economic shocks. Additionally, there are possible solutions to the problem of limited information, such as Factor Augmented VAR or dynamic rotations.

## Data Revisions and Real-Time Forecasting

At a given point in time, a forecaster will have access to data on macroeconomic variables that have been subject to different numbers of rounds of revisions, leading to varying degrees of data maturity. Observations referring to the very recent past will be first-release data, or data which has as yet been revised only a few times. Observations referring to a decade ago will typically have been subject to many rounds of revisions. How should the forecaster use the data to generate forecasts of the future? The conventional approach would be to estimate the forecasting model using the latest vintage of data available at that time, implicitly ignoring the differences in data maturity across observations. The conventional approach for real-time forecasting treats the data as given, that is, it ignores the fact that it will be revised. In some cases, the costs of this approach are point predictions and assessments of forecasting uncertainty that are less accurate than approaches to forecasting that explicitly allow for data revisions. There are several ways to “allow for data revisions,” including modeling the data revisions explicitly, an agnostic or reduced-form approach, and using only largely unrevised data. The choice of method partly depends on whether the aim is to forecast an earlier release or the fully revised values.