As a rule, the Shiller Index uses the CPI as reported for All Urban Consumers (CPIAUCNS on Fred(r)).
But the Index is only updated Quarterly, so monthly data is estimated. Which produces a very interesting difference over August, not to mention September expectations:
The annualised inflation rate between June and August is 0.39%, which just shows that the trend is volatile. But if there really is another round or two of 2.5% annual inflation, the projected 3% growth for Q3 is either going to look a lot more anemic than we think, or there is going to be a major producvtivity increase.
The odds of being paid the Marginal Product of Labor just dropped a bit more.