a measure of inflation [can be] defined by two statistical properties: (i) all prices increase in exactly the same proportion, and (ii) the change is unrelated to any relative-price movements. The extent to which (ii) holds is what we might call the measure’s 'purity'.

Their statistical approach is

[..] that factor analysis also gives a natural way to purify the measure of inflation. Factor analysis produces a set of components (or factors) that explain why prices move together. One of these factors is the equiproportional change in prices [..]. But the other factors are just as interesting. These factors are measures of relative-price changes due to some common source (say productivity, fiscal, or monetary shocks), and it turns out that a few of these alone account for a great deal of the variability of price changes. Therefore, we can use them to statistically purify our measure of inflation from these main sources of relative price movements.

Their results can be summarizes as follows:

Only around 15-20% of the movements in these measures of inflation correspond to pure inflation. Most of the time, pure inflation and CPI inflation are broadly related, but there are interesting exceptions. For instance in the late 1990s and early 2000s, CPI and core inflation were relatively low, but pure inflation was actually quite high. Favourable relative-price shocks seem to have accounted for most of what was seen back then as surprisingly low and stable inflation in spite of loose monetary policy.And,

What we found was reassuring. After controlling for relative price changes, the correlation between inflation (or pure inflation) and real activity is essentially zero. So, when we see that high inflation typically comes with low unemployment or high output, this is indeed driven by the change in relative prices hidden within the inflation measure. When there is pure inflation, that is when all prices increase in the same proportion independently from any relative price changes, nothing happens to quantities. Neoclassical economics seems to have this one right.Obviously, the theoretical background in their work is not well defined. They do not make any assumptions in the quantity-theory context about the only underlying reasons for absolute inflation, i.e. an increase in money, ; a relative price change cannot be translated into an absolute price increase with a constant money stock with given velocity and constant real output. What is interesting to me and especially to my co-blogger amv is that in times of high measured inflation and high economic activity, inflation always stems from relative price changes whereas the correlation between pure inflation and economic activity is almost zero (see the vertical Phillips curve). Well, this is what ABC-theory tells us as well as neoclassical theory.