We estimate the oil price pass-through into consumer prices both in the U.S. and in the euro area. In particular, we disentangle the specific effect that an oil price change might have on each disaggregate price, from the effect on all prices that an oil price change might have since it affects the whole economy. To do so, we first estimate a Dynamic Factor Model on a panel of disaggregate price indicators, and then we use VAR techniques to estimate the pass-through. Our results show that the oil price passes through core inflation only via its effect on the whole economy. This pass-through is estimated to be small. but statistically different from zero and long lasting.