Dear @cfp
I am second year ph.d student in South Korea.
Recently, I have been conducting stochastic simulation work using simple RANK model with Dynare OBC and found dynamics of price dispersion is somewhat weird.

In my knowledge, when zero lower bound on interest rate binds, price dispersion should rise significantly due to price rigidity. But stochastic simulation of OBC shows that price dispersion decreases when ZLB binds. Using second order approximation following Alves(2014,JME), I simulated series of implied price dispersion and these two series does not coincide in the times of ZLB (see file check 1).
\hat{v}_{t}^{p} = \frac{\varepsilon_{p}\overline{\theta}_{p}(\Pi-1)}{1-\theta_{p}}\hat{\pi}_{t} + \overline{\theta}_{p}\Pi \hat{v}_{t-1}^{p} + \frac{1}{2}\frac{\varepsilon_{p}\overline{\theta}_{p}(\varepsilon_{p}(\Pi-1)+1)(1-\overline{\theta}_{p}\Pi)}{(1-\overline{\theta}_{p})^{2}}\hat{\pi}_{t}^{2}

Using global method (GDSGE toolkit), I checked that price dispersion indeed rise in the times of ZLB (see file check 2).

I am using gurobi optimizer.
I tried several approach like including max operator in price dispersion dynamics, or writing mod file with above second order approximated process but dynamics of price dispersion exhibit same symptom.

If I am not missing something, could you help me with solving this issue ? I attached all the codes and figures. Your help will greatly aid doing my research.

Remember that DynareOBC is taking a first order approximation to the relative effect of the bound. So it’s not going to pick up (say) that severe ZLB episodes come with deflation which increases price dispersion. But I’m slightly surprised that you say DynareOBC is so different to a global approximation at the ZLB. In my experience, you don’t get so much deflation at the ZLB in plausible models, so dispersion can’t increase much if at all.

I also think that implied price dispersion dynamics (orange line in check 1) is somewhat too volatile meaning that deflation is extremely severe in the long ZLB episodes. I think price dispersion dynamics should be less volatile like simulation results using global methods (check 2 is simulation result of same model with same parameterization and using GDSGE toolkit).

From your answer, I think possible solution is modifying the model to generate less severe deflationary pressure in the long ZLB episodes. I think adopting more persistent interest rate smoothing or inflation indexation in NKPC may mitigate the discrepancy between two price dispersion dynamics. If possible, could you recommend any suggestion to handle this issue?

I have to evaluate E(\hat{v}_{t}^{p}) but using implied dynamics of 2nd order approximated price dispersion formula seems to be exaggerating the true mean.

Reason for using dynare OBC is that I have to simulate model with many different parameterization. Global method is feasible but requires too much computational burden and time…

My last question is the difference between order option in stochastic simulation. It seems using order = 2 option in dynare OBC is different to the using order = 2 in normal dynare. Could you explain what effect is captured in order=2 simulation compared to the order = 1 simulation in dynare OBC?

In the absence of OBCs DynareOBC gives the same second and third order solution as the NLMA toolkit of Meyer-Gohde. This is almost (but not exactly) the same as Dynare with pruning. For the OBC case, read my papers!