Demeaning with official targets

Hope all is well!
I’m going to estimate a model but two questions arose when I was transforming my data to the model equivalent. The model is an already log-linearized similar to the 3-basic equations of Galí but adapted to a small open economy.

  1. As of my knowledge, for the output gap, I need to filter (with a consistent filter such as the one-sided HP filter) the log of real GDP per capita. However, I have seen many published papers using the log of real GDP. Am I missing something or I shall use it in per capita terms?

  2. As this model is already log-linearized, the implied steady state is zero, so I need to demean all my observables such its mean is also zero. However, I’m struggling if I need to do this also for inflation and for interest rate. My concern is the following: in my country the inflation target is 3% on annual basis but the actual mean of inflation is closer to 4.5%. Initially, I would like the interest rate in the model to react to inflation above the 3% such as the Central Bank. However, doing this, my data is not consistent with the model because the implied mean in my data will be close to 1.5% (never hovering around the model’s steady state). Do you have some advice to tackle this issue?

As always, thanks in advance for your help!

  1. Yes, ideally you divide by population, but sometimes it is assumed that the population changes follow a slow-moving trend that is removed by the filter as well.
  2. Dealing with average levels of the nominal interest rate and inflation is tricky. The most common approach is to demean both the interest rate and the inflation and to then independently set the steady state inflation rate and the discount factor that governs the real interest rate to values that the modeler chooses. That way, the data won’t imply an restriction in this case. But obviously, you restrict the data to be in steady state on average. Regarding inflation target vs. realization: the steady state inflation rate need not be the same as the inflation target of the central bank.
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