## Wednesday, January 3, 2018

Some years ago I had predicted that Canada would begin to undershoot its 2% inflation target, and then touted the success of the information transfer monetary model when that prediction came true. However I mostly see the monetary model as at best a local approximation with the dynamic information equilibrium model being better empirically (discussion in terms of US inflation at this linked post).

To that end, I thought I'd put together how you'd look at Canada's below-target inflation in terms of the dynamic information equilibrium model (of all items CPI). In this case, the dynamic equilibrium is approximately 2%, and the undershooting is due to a long-duration shock possibly triggered by the global financial crisis/Great Recession.

The first graph is the full CPI level dataset from FRED. The second shows a more recent CPI level data. The third shows year-over-year inflation. The main shocks are the demographic shock centered at 1978.65 ± 0.04 (width [1] = 3.0 y) and the post-crisis shock is centered at 2017.7 ± 4.9, with a width of 3.3 years. There are two additional shocks in 1991 and 1993 to deal with the bump in the CPI.

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Footnotes

[1] I've been a bit sloppy on this blog about what I mean by the "width" of a transition, although I nearly always use the "width" or "inverse steepness" parameter $b_{0}$ of the logistic function

$$f(t) = \frac{a_{0}}{1+e^{-\frac{t-t_{0}}{b_{0}}}}$$

Since the derivative is nearly a Gaussian function, we can think of the 1-standard deviation width $\sigma$, which is approximately

$$\sigma \approx \sqrt{\frac{8}{\pi}} b_{0} \simeq 1.6 b_{0}$$

based on matching the leading order of the Taylor series. The other possible measure is the full width at half maximum ($FWHM$) which is

$$FWHM = 2 b_{0} \log \left(3 + 2\sqrt{2} \right) \simeq 3.5 b_{0}$$

Therefore if $b_{0} \simeq 3.0\;\text{y}$ means $\sigma \simeq 4.8\;\text{y}$, and $FWHM \simeq 10.6 \;\text{y}$. Using the $\sigma$ measure, 95% of the shock occurs within $4 \sigma$ distances (i.e. $2 \sigma$ on either side) or 19.1 years.