Updated to include core PPI:

Figure 1: Instantaneous inflation for core CPI (blue), chained core CPI (tan), supercore CPI (pink), services supercore (light green), PPI core (red), and PCE core (green), per Eeckhout (2023), T=12, a=4. September observation based on PCE core nowcasts from Cleveland Fed as of 10/10/2024. Source: BLS, BEA, Pawel Skrzypczynski, Cleveland Fed, and author’s calculations.

Instantaneous Core PPI inflation declined. It’s tempting to conclude this movement will feed into the CPI inflation, but this is not obviously the case. See this post for discussion of formal econometric analyses of whether PPI leads CPI.