All data for this post is from Shiller's website (link in the post). I used this data because it's a canonical data set used in many other analyses. His price data is monthly, all prices are 'real' i.e. inflation adjusted, dividends are not included, however his return data does include dividends i.e. it's 'total return'. It was his choice to use ten years for forward returns.

I hope I was clear that the FAPE1 and FAPE2 "alternatives" are in fact straw men that I set up as experiments, fully expecting to knock them down.
But to my dismay, they didn't fall down!

FAPE1 substitutes for the earnings data, i.e. the E in CAPE, the values from the trend line of earnings over time. The trend line is completeley smooth, it has no features or wiggles of any sort. But FAPE1 still looks like CAPE (the graph) and feels like CAPE (the accuracy).

FAPE2 doesn't use earnings at all. But it still looks like CAPE (the graph) and feels like CAPE (the accuracy). To answer a question, I first tried 10 years for the average in FAPE2 but doubled it to 20 years because 20 worked better, that's the only tuning. IMO, FAPE2 is both amusing and somewhat alarming, but a bit of a trick. It's FAPE1 that I find more disconcerting: a simple change to CAPE i.e. remove all details about earnings other than "earnings go up" and it still works.

In some initial investigations I tried the 'Hull average', because it's supposed to have less lag than the SMA that Shiller used. But "solving lag" turned out not to be the point at all. In fact, you can make lag worse by artificially lagging Shiller's earning data by years, and not affect the outcome of CAPE very much.

That point is that CAPE is surprisingly and distressingly insensitive to details of earnings: you can make the earnings totally smooth as in FAPE1, you can even totally eliminate use of earnings as in FAPE2, and it'll still look like CAPE (the graph) and have the feel of CAPE (similar accuracy).

I had thought, as many others do (lots of papers are written on CAPE), that it's the interplay of details of E and P in Shiller's CAPE that makes CAPE predictive and that CAPE is a shining example that "valuation matters".
But you can quash, and even kill (eliminate) the details of E, and "CAPE" (i.e. the FAPE1 and FAPE2 versions of CAPE) still work.
To be clear, I'm not saying that valuation doesn't matter (I believe that it does). I'm saying that CAPE doesn't show that valuation matters.