Fixing a variable across leads

General DSGE Modeling
1

I am trying to define a variable ( let’s say x ) such that it always equals another variable ( let’s say y ) but not only today but also in all future periods, i.e. x(+k) = y for all k >= 0. Conceptually, I want to “fix” x at time t so that it does not change its value across leads. To be even more specific, I need the following equations in Dynare:

1. x = y;
2. x(+1) = y/PI(+1);
3. x(+2) = y/(PI(+1)PI(+2));
…
k. x(+k) = y/(PI(+1)PI(+2)*…*PI(+k));

In Dynare, I would like to write all these equations. However, I cannot add two or more equations to define one ondegenous variable (that is, x) because the system would be over-identified. Is there a way to spell out the idea reported above without overidentifying the model?

2

The equations you define there are forward looking, i.e., x(+2) = y/(PI(+1)PI(+2)); means

E_t x_{t+2} = E_t\frac{y_t}{\Pi_{t+1}\Pi_{t+2}}

is that really what you want?

3

Correct, for all k. Many thanks!

4

This looks like you are trying to bypass rational expectations by providing a value for the k-step ahead expectations without it following out of your model equations. That won’t work in Dynare. Usually, you have a process for x_t that results in E_{t}x_{t+k} following the indicated process.

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