Set identification in Bayesian vector autoregression (VARs) is becoming increasingly popular while facing recent criticism about potentially unwanted prior dominance and underrepresented bounds of the identified set. This can lead to biased inference even in large samples. Common estimation strategies in high dimensions or with tight restrictions can prove to be highly inefficient or even practically infeasible. We propose to include micro data on heterogeneous entities for the estimation and identification of vector autoregressions to achieve sharper inference. First, we provide conditions when imposing a simple ranking of impulse responses will sharpen inference in bivariate and trivariate VARS. Importantly, we show that this sharpening also applies to variables not subject to ranking restrictions. Second, we develop two types of inference to address recent criticism: (i) A prior-robust posterior over the bounds of the identified set and (ii) a fully Bayesian sampling algorithm that allows us to efficiently include an agnostic prior over the non-identifiable parameters. Third, we apply our methodology to US data to identify productivity news and defense spending shocks. We find that under both algorithms the bounds of the identified sets shrink substantially under heterogeneity restrictions relative to standard sign restrictions.
Bayesian VAR sign restrictions set identification micro data news shocks defense spending