Common Drifting Volatility in Large Bayesian VARs

The general pattern of estimated volatilities of macroeconomic and financial variables is often broadly similar. We propose two models in which conditional volatilities feature comovement and study them using U.S. macroeconomic data. The first model specifies the conditional volatilities as driven by a single common unobserved factor, plus an idiosyncratic component. We label this model BVAR with general factor stochastic volatility (BVAR-GFSV) and we show that the loss in terms of marginal likelihood from assuming a common factor for volatility is moderate. The second model, which we label BVAR with common stochastic volatility (BVAR-CSV), is a special case of the BVAR-GFSV in which the idiosyncratic component is eliminated and the loadings to the factor are set to 1 for all the conditional volatilities. Such restrictions permit a convenient Kronecker structure for the posterior variance of the VAR coefficients, which in turn permits estimating the model even with large datasets. While perhaps misspecified, the BVAR-CSV model is strongly supported by the data when compared against standard homoscedastic BVARs, and it can produce relatively good point and density forecasts by taking advantage of the information contained in large datasets.

宏观经济与金融变量的估算波动率通常呈现大体相似的一般模式。我们提出两类具备条件波动率协同变动特征的模型，并采用美国宏观经济数据开展相关研究。首个模型将条件波动率设定为受单一公共不可观测因子驱动，并附带特质成分。我们将该模型命名为含通用因子随机波动率的贝叶斯向量自回归（Bayesian Vector Autoregression, BVAR-GFSV），研究表明，假设波动率存在公共因子所带来的边际似然损失相对温和。第二款模型我们命名为含公共随机波动率的贝叶斯向量自回归（BVAR-CSV），它是BVAR-GFSV的特例：移除特质成分，并将所有条件波动率对公共因子的载荷系数统一设定为1。此类限制为VAR系数的后验方差提供了便捷的克罗内克（Kronecker）结构，进而即便面对大规模数据集，也可实现模型的估计。尽管该模型可能存在设定偏误，但相较于标准同方差BVAR模型，BVAR-CSV模型仍得到了数据的强力支持；此外，通过充分利用大规模数据集蕴含的信息，该模型可生成质量较为出色的点预测与密度预测。