When considering aggregate inflation it is generally difficult to know whether movements stem from broad based price changes or are confined to specific components.
Analysts often attempt to overcome this issue by considering price changes of major components and making inferences about the impact of these on the aggregate inflation rate. However, these inferences are imprecise and do not provide a complete accounting of aggregate inflation. A much more precise picture is can be achieved by decomposition of the inflation rate.
Decompositing aggregate inflation rates mean reverse engineering the index construction and the methodology hence depends on the specific index. Below is a brief account for how the decomposition is done for different indexes.
The HICP indexes are genrally based on Lasperey indexes with annual chain linking, see ECB (2000). The annual chain-linking implies that any decoposition will have to be divided into two components.
Some european countries have a slightly more elaborated setup with non standard updates of weights. For now, these breakdowns are not calculated.
The PCE index is an Ficher ideal chain index, with chain linking between all period, see BEA. The breakdown formula follows Bauer et al. (2004).