I would like to replicate in dynare the paper by Del Negro, Giannoni and Schorfheide: Inflation in the Great Recession and New Keynesian Models:
newyorkfed.org/research/staf … /sr618.pdf
They observe 10-year inflation expectations. To avoid to deal with a large state vector, they first solve the model without inflation expectations in the measurement equation. Then they compute expected inflation 40-quarter ahead using the solution of the model. Finally they append the additional measurement equation (eq 32) relating data and model inflation expectations. I wanted to know whether it is possible to do something similar in dynare, or I need to have inflation(+40) in the mod file, which I suppose would make the solution and estimation substantially slower.
I may be wrong here, but the inflation term here is purely forward looking. Using inflation(+40) should not add any states to the model. The less in speed should hence be manageable. Moreover, I don’t think Del Negro, Giannoni and Schorfheide actually first solve the model without inflation expectations. Rather, they solve it with inflation expectation and then show how to link those inflation expectations to the data. Thus, you might not get around specifying the forward-looking terms. Their second line of writing equation (32) seems like what Dynare would do if you enter the first line into the model. But I have not tried to derive this second line.
I think the difference is that if you write the definition of inflation expectations in the model block (the average inflation expectations ‘piexp’ from 1 to 40), Dynare needs to create 39 auxiliary variables. Instead, what they do is to use the solution of the model (without piexp) to compute inflation expecations. You are right that the number of states is the same in both cases, what differs is the muber of jumps (12 vs 51). It seems a nice trick, especially if one wants to introduce additional observations reflecting average expectations. But maybe your point is that the difference in speed is minimal.
I am not sure I understand the point. I don’t see how you can use the model solution without solving the model. I read this section to state that the solution used to get from the first to the second line is the one of the full model. It does not read as if the solution used for that equation is the solution to a different kind of model that does not feature this inflation expectations term.
They first solve the model without measurement equations in the model block. Then they construct the measurement equations separately. As you point out, they can do it exactly because the measurement equations do not require additional state variables. To construct the average inflation expectations you simply need to cumulate inflation 1 to 40 periods ahead using the transition equation (the second line in equation 32).
The less parsimonious way, which is what I currently do in dynare, is to add the measurement equations directly in the model block, which lead to the creation of 39 jump variables (as inf(+1) is already in the model). I guess in dynare those two approaches might imply similar speeds because I am just adding jump variables?
I guess the anwer to my question is that dynare uses a Schur decomposition to solve for the transition equation for the state variables only. So effectively dynare already does what DGS do.