As a follow-up to this post, I show what the graphs look like if I use M1 for the US instead of the St. Louis adjusted monetary base (left is the normalized sigma-kappa space and the right are the fits to the price level):
Note that M1 and the monetary base are different things: the former measures currency + deposits outside the Fed and the latter measures currency + deposits at the Fed. I had previously used M1 as a proxy for the base (when I couldn't find proper data) because they are at least of similar magnitude for the US (as opposed to say M1 and M2), but this shows that prescription leads to quite different results. Altogether, using M1 looks like a mess.
And since I found monetary base data for Japan, here is what the US and Japan data look like in terms of monetary base:inflation data for the US relatively well; for Japan there are still some wiggles. I'd venture to say that inflation measures things over a longer average period than the relatively instantaneous effect in this model. Here are the fits (empirical data = dashed curves, model fit = solid curves):