We introduce a new class of time-varying parameter vector autoregressions (TVP-VARs) where the identified structural innovations are allowed to influence the dynamics of the coefficients in these models. An estimation algorithm and a parametrization conducive to model comparison are also provided. We apply our framework to the US economy. Scenario analysis suggests that, once accounting for the influence of structural shocks on the autoregressive coefficients, the effects of monetary policy on economic activity are larger and more persistent than in an otherwise standard TVP-VAR. Our results also indicate that cost-push shocks play a prominent role in understanding historical changes in inflation-gap persistence.