When I made the discovery (or, corrected the error) that only the currency component of the monetary base ("M0", rather than MB) was used to determine the price level I was initially pretty excited: it improved the fit to price level data substantially after 2008. Lurking unmentioned in the background, however, was a terrible blow. Using M0 meant that one of the most interesting aspects of the information transfer model would have to be scrapped: the path of the economy, interest rates and the price level surface could no longer be combined into a single plot in (MB, NGDP) space that enabled one-plot economic situational awareness.
In less than 24 hours, the sadness has been replaced by excitement, and I feel like the failure has lead to new understanding. Let's start with the new 3D graph of the price level surface using just the currency component of the base:
We can see that the US economy is on the good side of the information trap criterion (dark line where ∂P/∂MB = 0) as opposed to having crossed it (see e.g. here, which leads to deflationary monetary expansion). It is still close enough to the line that liquidity trap-like conditions are present. This graph forms the basis of the (MB, NGDP)-space diagram, but first we need to address interest rates.
Previously, the monetary base including reserves (i.e. the "QE" component) was used to fit short term interest rates rather well (for several countries). "M0" doesn't work as well with the 3-month rate. What could be the difference? Scott Sumner frequently likes to point out that things like QE boosting central bank reserves are seen as temporary, whereas printing currency would be seen as more permanent. This gave me an idea: what if the currency component of the base controlled long term interest rates while currency + reserves controlled short term rates? Let's say we have two markets rl:NGDP→M0 and rs:NGDP→MB with the same information transfer index (κ) where rl is the long run interest rate (10 year treasury rate) and rs is the short run interest rate (3 month rate). I fit κ to the 10 year treasury rate rl:NGDP→M0 and then looked at how well the MB data fit the 3-month rate in the same function. The result was pretty amazing:
The M0 model result is the darker blue line, while the MB result is the lighter blue one. The 10 year rate is darker green, while the 3-month rate is lighter green. The fit for both uses the function log r = c log (1/κ) (NGDP/Mx) with κ = 10.4 and c = 2.8 [update: corrected equation to have c on the RHS], and Mx being either the currency component ("M0") or currency + reserves (MB).
When I saw this plot, I had that great feeling that I've discovered something! Now it doesn't give you precise enough results to make trades (note that it's a log scale on the vertical axis), but I believe this illuminates something fundamental about interest rates. It also enables me to make that (MB, NGDP)-space diagram! Since both P(M0, NGDP) and r(M0, NGDP) are both functions of the same M again (due to fitting the long run rate to M0), we can plot the level curves of the price level and the interest rate on the same graph. Below you can see the "M0" component as a dark blue line, the MB as the light blue one, the interest rate level curves as red lines and the price level level curves as gray lines (the ∂P/∂MB = 0 line is shown as a dark line):
We can see that the long run interest rate is above the "liquidity trap" rate (it hasn't crossed the black line), but the short run rate is below it -- further quantitative easing (QE) would likely not stimulate the economy. However, printing currency could lower long term rates and increase inflation.