Dennis Fixler and Ted Jaditz (2002) "An Examination of the Difference Between the CPI and the PCE Deflator."
Both the Bureau of Labor Statistics (BLS) and the Bureau of Economic Analysis (BEA) produce measures of the change in prices the consumers pay on the goods that they consume. While these measures tend to agree in broad historical trends in prices, they sometimes give different pictures of inflation over short horizons.
There are several reasons why these indexes differ. First, the two indexes use different formulas. For the period in question, the CPI is a Laspeyres index, while the BEA product is a Fisher Ideal index. Second, the two indexes have different underlying concepts. The BLS product measures the prices paid by (urban) consumers, while the BEA product measures the prices of final consumption goods, wherever they are purchased. Finally, even when there is significant agreement across indexes in the broad outlines of coverage, differences in how the detailed components are implemented lead to differences in how prices are measured and differences in the weights attached to specific indexes.
We quantify the magnitude of each of these factors. There is no one "smoking gun" that explains the discrepancy between the indexes. Rather, the overall discrepancy is the result of the accumulation of a number of small effects.