This paper uses factor-augmented vector autoregressions (FAVAR) estimated using a large data set to disentangle fluctuations in disaggregated consumer and producer prices which are due to macroeconomic factors from those due to sectorial conditions. This allows us to provide consistent estimates of the effects of US monetary policy on disaggregated prices. While sectorial prices respond quickly to sector-specific shocks, we find that for a large number of price series, there is a significant delay in the response of prices to monetary policy shocks. In addition, price responses display little evidence of a 'price puzzle,' contrary to existing studies based on traditional VARs. The observed dispersion in the reaction of producer prices is relatively well explained by the degree of market power, as predicted by models with monopolistic competition.