Macroeconomists often estimate impulse response functions using external instruments (proxy variables) for the shocks of interest. However, existing methods are silent on the importance of the structural shocks for macroeconomic fluctuations. We provide tools for doing inference on variance decompositions in a general semiparametric moving average model, disciplined only by the availability of external instruments. The share of the variance that can be attributed to a shock is partially identified, albeit with informative bounds. Point identification of most parameters, including historical decompositions, can be achieved under additional assumptions that are weaker than invertibility, a condition imposed in conventional Structural Vector Autoregressive analysis. In fact, we prove that invertibility is testable in the presence of external instruments. We illustrate the practical usefulness of our methods by obtaining a tight upper bound on the importance of monetary policy shocks for U.S. inflation dynamics.