Hi Nicolas,
Interesting idea. One critical point is, as your article points out PE returns are an estimate, not an observation. The CA index uses the PME methodology for calculating "PE" returns, the math of its calculation can make any time series appear correlated with the stock market, or a subset by picking the appropriate public market "discount rates". Using the PME approach, one can make e.g. the sale of Taylor Swift records appear highly correlated to the stock market. The point is that PME should be used and compared to with high caution.
I was wondering how this inherent bias is treated in your work.
Best.