Livio Di Matteo has a post over at Worthwhile Canadian Initiative about interest rates in Canada over the long run. In it he posits the idea that the ratio of the money supply to GDP is correlated with interest rates and shows an inverse correlation between M2/GDP and the Canadian bank rate.
Constructing the IS-LM model in the information transfer model gives us an interest rate model that is a ratio of NGDP to the money supply (in our case the monetary base MB). Our market is transferring information from the aggregate demand to the monetary base that is detected by the interest rate, or in the notation I have been using r:NGDP→MB. This implies that r ~ NGDP/MB (according to the "first law" of information transfer economics) which is (the inverse of ) the ratio given Di Matteo .
I had already looked at the model for Canada awhile ago; I was hoping to use the extremely long run data Di Matteo points to in his post however I still don't have GDP, CPI or interest rate data that goes back to the 1870s. I did manage to expand the domain of the model from 1980-2009 back 1960s so here are the results for the price level and interest rate:
And here we can see the correlation between the model ratio and the interest rate:
Actually, the model for Canada seems to work better as two separate models for before the 1980s and after which makes me think that my maybe I should think of the information transfer model as a local approximation to the true underlying model:
Local approximations are a more conservative interpretation of these kinds results than e.g. phase transitions or monetary policy regimes, but if I didn't make bold claims from time to time this blog would be even less interesting.
PS I agree with Di Matteo's conclusion that low interest rates will persist for a long time.
 Since Canada has low information transfer index (κ ~ 0.6), M1, M2 and the MB are all roughly parallel which means that the difference is basically a scale factor.