Here are Ricardo Reis and Mark Watson who will tell you what they have done and why it matters. This is interesting work:
Relative Goods' Prices and Pure Inflation, by Ricardo Reis and Mark W. Watson, NBER WP 13615, November 2007 [open link]: ABSTRACT This paper uses a dynamic factor model for the quarterly changes in consumption goods' prices to separate them into three components: idiosyncratic relative-price changes, aggregate relative-price changes, and changes in the unit of account. The model identifies a measure of "pure" inflation: the common component in goods' inflation rates that has an equiproportional effect on all prices and is uncorrelated with relative price changes at all dates. The estimates of pure inflation and of the aggregate relative-price components allow us to re-examine three classic macro-correlations. First, we find that pure inflation accounts for 15-20% of the variability in overall inflation, so that most changes in inflation are associated with changes in goods' relative prices. Second, we find that the Phillips correlation between inflation and measures of real activity essentially disappears once we control for goods' relative-price changes. Third, we find that, at business-cycle frequencies, the correlation between inflation and money is close to zero, while the correlation with nominal interest rates is around 0.5, confirming previous findings on the link between monetary policy and inflation.
6. What have we done and why does it matter? In this paper, we ... used different estimation techniques and specifications to robustly estimate pure inflation, and proposed a simple method to compute macroeconomic correlations while controlling for goods’ relative price changes.
Our first finding was that pure inflation can differ markedly from other conventional measures of inflation, like the PCE deflator or its core version. It is smoother, less volatile, and in particular in the 1990s, its ups-and-downs are quite different from those in other measures of inflation. This should be useful to economic historians since it provides an alternative account of the movements in inflation in the last half-century. Relative to existing measure of inflation, pure inflation has the virtue of separating absolute from relative-price changes, which is a crucial distinction in economic theory. Moreover, pure inflation matches more closely the concept that many economists seem to have in mind when discussing aggregate movements in prices and monetary policy (typically based on intuition that comes from a one-good world).
Our second main finding was that pure inflation was quantitatively significant (it accounts for about 5% of individual price changes), but only accounts for 15-20% of the variability in inflation measured by conventional price indices, like the PCE deflator, the GDP deflator, or the CPI. This has at least two implications for the work of economic theorists building models to explain inflation. First, it shows that comparing the predictions of one-good models with common measures of inflation is flawed. The difference between these measures and pure inflation is large enough that it can easily lead to mistakenly accepting or rejecting models. Second, our estimates provide a new test statistic with which to test the pricing assumptions of models with many goods. An important ingredient (and topic of debate) in recent models of nominal rigidities is a model of pricing that implies slow adjustment of prices to monetary policy shocks together with frequent price changes. The fraction of the variability of cost-of-living inflation accounted for by pure inflation can be an important statistic to diagnose the success of these models at fitting the data.
Third, we found that, once we controlled for relative goods’ prices, the Phillips correlation became quantitatively insignificant. Therefore, the correlation between real quantity variables and nominal inflation variables that we observe in the data can be accounted for by changes in relative prices. This implies that models that break the classical dichotomy via nominal rigidities in good’s price adjustment are likely more promising than models that rely on money illusion on the part of agents.
Fourth, we found that pure inflation is partly related to monetary policy variables. The link to the growth rate in monetary aggregates is weak, but the correlation with nominal interest rates at business cycle frequencies is strong (approximately 0.5).
To conclude, economic theories have strong predictions on whether and when there should be pure inflation and what its effects would be, and discussions of monetary policy often revolve around its relation with pure inflation. However, observing pure inflation is naturally difficult, since the concept itself is more a fruit of thought experiments than something easily observed. As a result, there have been few systematic attempts to measure it in the data. The goal of this paper was to make some progress on measuring pure inflation and understanding its effects. Our estimates are certainly not perfect. We hope, however, that they are sufficiently accurate that future research can look deeper into the time-series and the moments that we provide, and that by stating the challenges and putting forward a benchmark, we can motivate future research to come up with better estimators. Likewise, we are sure that our findings will not settle the debates around the key macroeconomic correlations. Our more modest hope is that they offer a new perspective on how to bring data to bear on these long-standing questions.