Average inflation targeting

Hi APK,

Suppose time in your model is a quarter, as it usually is.
Quarterly inflation is the defined as the time difference in the log of the price level, such that simply \pi_t=p_t - p_{t-1}. By the same token, annual (4-quarter) inflation is \pi^{(4)}_t=p_t - p_{t-4}.
Now, suppose you would like to have a 3-year averaging window.
The 3-year (12-quarter) average inflation rate (at an annual rate) in quarter t, \bar{\pi}^{(12)}_t is defined as \bar{\pi}^{(12)}_t=\frac{p_t - p_{t-12}}{3}.

You could implement the definitions in the code by writing

plevel     = plevel(-1)+pinf;
pinf3yrs  = (plevel-plevel(-12))/3;

With such definitions in mind, you can write the Taylor rule along the lines
of \left(\ldots\left(\frac{\bar{\pi}^{(12)}_t}{\bar{\pi}^{\star}}\right)^{\phi_\pi} \right)^{1-\phi_r}, where the dots denote any other term you may want to include in your Taylor rule and \pi^{\star} is your objective for the 3-year average inflation rate (at annual rate), say 2%.
In the code you could write it as

r = ... (...(pinf3yrs/pi_star)^phi_pi)^(1-phi_r)

You can then optimize over \phi_\pi as usual. :grinning:

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