In Smets and Wouters 2007, they use dlP_t = \overline{\pi} + \pi_t as the measurement equation for the inflation where \overline{\pi} = 100(\Pi^*-1) which is net inflation rate in percentage terms. On the contrary, LHS of the equation is the gross inflation rate. How can we have gross rate in one side and net rate on the other side?

Due to \ln(1+x)\approx x for small x, the logarithm of the gross inflation rate is approximately equal to the net inflation rate. That’s what’s used on the left.

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Thank you so much Prof. Pfeifer