I am trying to replicate the results in Andrea Ferrero’s paper

ecb.europa.eu/pub/pdf/scpwps/ecbwp502.pdf

However, I do not know how to write the following which can be found on page 53,

I think I know how to write the infinite sum using a macro processor but I am not sure about two things:

-why there is an infinity in the first place if the whole thing is log-linearized

-how do I write a sum in terms of t, the period we are in, in Dynare?

Thank you very much,

Log linearization does not convert infinite sums to finite sums. The trick rather is that log-linearization often makes it easier to find a recursive representation that gets rid of the infinite sum. That is what is done on that page 53. After

follows a recursive equation in v_{F,t}. This equation should completely determine the evolution of v_{F,t}. This variable v_{F,t} is used to get rid of the variable v_{F,s} that only has a representation as an infinite sum and should not appear in any other equation in the model and can thus be left out when entering the model. In a nutshell, the recursive definition of v_{F,t} replaces v_{F,s}.

Thank you very much, that was really helpful.