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Public abstract
We develop a unified framework that combines shock volatility with sign and narrative restrictions and provides the theoretical foundation for the computationally efficient sampler HARS. HARS preserves the heteroskedastic likelihood and can be combined with any posterior simulator for the heteroskedastic model. In monetary policy, oil market, and fiscal policy models, the same restrictions deliver substantively different economics once shock heteroskedasticity is accounted for. Homoskedastic SVARs put uncertainty in the wrong place, pushing shock-scale variation into impulse-response uncertainty. Heteroskedasticity sharpens dynamic responses, alters economic conclusions, and restores 90% credible intervals as a practical standard for economic inference.