Issues with a non-stationary New Keynesian model

Dear all,

I am having difficulties trying to solve the model of Barsky & Sims (2012, AER, Information, Animal Spirits and the Meaning of Innovations to Consumer Confidence). Their code is available on MATLAB, but I made an effort to write it in a Dynare file.

There is one thing I am struggling with: their model has a technology process (log) which follows an unit root with drift process, the drift being the growth rate of technology (log), which itself follows an AR(1) process. The authors use the unit root equation “a = a(-1) + ga(-1) + ea” when solving the model by the Anderson & Moore algorithm (AIM) as well as the log of technology in other equations like “y = a – alpha*k(-1) + (1-alpha)*n”. Both MATLAB and Dynare solve the model and return the same results. My question is: should the model not be stationarized first?

I would really appreciate some light regarding this question.
Thank you all.

PS: This is just the part where they solve the full-information model, and I ignored the flexible price equations. Codes below. I tried writing a stationary version of it too, but for whatever reason it does not satisfy Blanchard-Khan conditions. I checked the math a few times but do not seem to find the error.
Test_BS_stationary.mod (3.01 KB)
Test_AIM.m (15.1 KB)
Test_BS.mod (1.81 KB)

As long as you do simulations, unit roots are not an issue (the BK conditions allow the root on the unit circle). In the absence of shocks, the model would be stationary.
It would only be problematic if the drift had a constant part, giving rise to trend growth. In that case, variables would explode even without shocks.

Estimation would be an issue, but that is not relevant.

Thank you very much for the helpful reply, Professor.

However, I forgot to mention the drift has indeed a constant part, as is defined in the paper: “ga = (1-rho_ga)g + rhogaga(-1) + ega”. I am not sure why this term does not appear on the MATLAB (hence, also on my replication within Dynare) code. The only reason I imagine it disappearing is expressing ‘ga’ as deviations from steady state, but then would it not be necessary to express ‘a’ as deviations from SS as well? (Which I believe would be problematic since ‘a’ does not have a steady state).

Also, later on the distance between IRF’s from the data and the simulated model are minimized. Would this be an issue in the sense of estimation you mentioned?

I thank you again for your time, Professor.

It seems they detrended by the deterministic growth part, which is the constant g. When you do this, everything can be interpreted as deviations from this long-run deterministic trend.

And no, IRF matching is not affected by the estimation problems, because you consider a one-time shock in the IRFs. Therefore, you should be fine.