Vector autoregressions (VARs) are flexible time series models that can capture complex dynamic interrelationships among macroeconomic variables. However, their dense parameterization leads to unstable inference and inaccurate out-ofsample forecasts, particularly for models with many variables. A solution to this problem is to use informative priors, in order to shrink the richly parameterized unrestricted model towards a parsimonious naïve benchmark, and thus reduce estimation uncertainty. This paper studies the optimal choice of the informativeness of these priors, which we treat as additional parameters, in the spirit of hierarchical modeling. This approach is theoretically grounded, easy to implement, and greatly reduces the number and importance of subjective choices in the setting of the prior. Moreover, it performs very well both in terms of out-of-sample forecasting-as well as factor models-and accuracy in the estimation of impulse response functions. Non-Technical SummaryVector autoregressions (VARs) are flexible time series models that can capture complex dynamic interrelationships among macroeconomic variables. However, their dense parameterization leads to unstable inference and inaccurate out-of-sample forecasts, particularly for models with many variables. With flat priors and conditioning on the initial observations, the posterior distribution of the VAR coefficients is centered at the Ordinary Least Square (OLS) estimate of the coefficients and it is easy to compute. It is well known, however, that working with flat priors leads to inadmissible estimators (Stein, 1956) and yields poor inference, particularly in large dimensional systems (see, for example, Sims, 1980;Litterman, 1986;Bańbura, Giannone, and Reichlin, 2010;Koop and Korobilis, 2010). One typical symptom of this problem is the fact that these models generate inaccurate out-of-sample predictions, due to the large estimation uncertainty of the parameters.In order to improve the forecasting performance of VAR models, Litterman (1980) and Doan, Litterman, and Sims (1984) have proposed to combine the likelihood function with some informative prior distributions. Using the frequentist terminology, these priors are successful because they effectively reduce the estimation error, while generating only relatively small biases in the estimates of the parameters.The literature has proposed a number of heuristic methodologies to set the informativeness of the prior distribution. In the context of VARs, for example, Litterman (1980) and Doan, Litterman, and Sims (1984) set the tightness of the prior by maximizing the out-of-sample forecasting performance of the model. Bańbura, Giannone, and Reichlin (2010) propose instead to control for over-fitting by choosing the shrinkage parameters that yield a desired in-sample fit. A number of papers have subsequently followed either the first (e.g. Robertson and Tallman, 1999;Wright, 2009;Giannone, Lenza, Momferatou, and Onorante, 2010) or the second strategy (e.g. Giannone, Lenza, and Reichlin, 2008...

Vector autoregressions (VARs) are flexible time series models that can capture complex dynamic interrelationships among macroeconomic variables. However, their dense parameterization leads to unstable inference and inaccurate out-ofsample forecasts, particularly for models with many variables. A solution to this problem is to use informative priors, in order to shrink the richly parameterized unrestricted model towards a parsimonious naïve benchmark, and thus reduce estimation uncertainty. This paper studies the optimal choice of the informativeness of these priors, which we treat as additional parameters, in the spirit of hierarchical modeling. This approach is theoretically grounded, easy to implement, and greatly reduces the number and importance of subjective choices in the setting of the prior. Moreover, it performs very well both in terms of out-of-sample forecasting-as well as factor models-and accuracy in the estimation of impulse response functions. Non-Technical SummaryVector autoregressions (VARs) are flexible time series models that can capture complex dynamic interrelationships among macroeconomic variables. However, their dense parameterization leads to unstable inference and inaccurate out-of-sample forecasts, particularly for models with many variables. With flat priors and conditioning on the initial observations, the posterior distribution of the VAR coefficients is centered at the Ordinary Least Square (OLS) estimate of the coefficients and it is easy to compute. It is well known, however, that working with flat priors leads to inadmissible estimators (Stein, 1956) and yields poor inference, particularly in large dimensional systems (see, for example, Sims, 1980;Litterman, 1986;Bańbura, Giannone, and Reichlin, 2010;Koop and Korobilis, 2010). One typical symptom of this problem is the fact that these models generate inaccurate out-of-sample predictions, due to the large estimation uncertainty of the parameters.In order to improve the forecasting performance of VAR models, Litterman (1980) and Doan, Litterman, and Sims (1984) have proposed to combine the likelihood function with some informative prior distributions. Using the frequentist terminology, these priors are successful because they effectively reduce the estimation error, while generating only relatively small biases in the estimates of the parameters.The literature has proposed a number of heuristic methodologies to set the informativeness of the prior distribution. In the context of VARs, for example, Litterman (1980) and Doan, Litterman, and Sims (1984) set the tightness of the prior by maximizing the out-of-sample forecasting performance of the model. Bańbura, Giannone, and Reichlin (2010) propose instead to control for over-fitting by choosing the shrinkage parameters that yield a desired in-sample fit. A number of papers have subsequently followed either the first (e.g. Robertson and Tallman, 1999;Wright, 2009;Giannone, Lenza, Momferatou, and Onorante, 2010) or the second strategy (e.g. Giannone, Lenza, and Reichlin, 2008...

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