Expectation

[danicara]

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

https://www.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,

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E_{t}[sum_{s=t}^{\infty} (\beta)^{s-t} v_{F,s}]

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,

[jpfeifer]

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

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Up to the first order, the flow version of the government budget constraint is

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}.

[danicaratelli]

Thank you very much, that was really helpful.