Following this post, I tried to look at some longer run data. In this case I used GNP (and GNP deflator, but still the Adjusted Monetary Base) from FRED for 1929 to the present and plotted two surfaces. One surface (white, blue curve) uses κ∼logMB/logGNP, the second surface (red) uses a constant value of κ=0.65. The result for the price level fit does well with varying κ (blue line) post WWII, while the constant κ (red dashed line) does worse post WWII, but much better with the earlier data (data is the green line)
Here are the surface plots:
The red surface for constant κ=0.65 follows the green line on average, while the blue curve (model, white surface) really messes up on the pre-WWII data. Why? I am not sure. The gold standard? Bad data?
One possibility is that for the early data, one or both of the constants c1,c2 become important ...
κ=logMB+c1logNGDP+c2