re vs deterministic simulation

[dynapie]

Hi,
I am simulating a model under rational expectations, using the stochsimul option: I am assuming a very persistent shock (say, 0.99 coeff of autoregression over 100 periods). How do I specificy this same persistent shock in a deterministic simulation (ie using the simul option) for the same number of periods? Will the results differ qualitatively, even if I assume a very long horizon (say 400) periods?

Thanks a lot,

Pietro

[MichelJuillard]

Don't confuse the model and the number of periods. In your model, you specify an autocorrelated shock with AR coefficient = 0.99. The innovation is declared as exogenous variable and the autocorrelated process itself is an endogenous variable. This is your model and you can set it in a deterministic or a stochastic context.

Let's assume that you consider a shock to the innovation in period 1. In a deterministic context, agents will take their decisions knowing that future value of the innovations will be zero in all future periods. In a stochastic context, agents will take their decisions knowing that future value of innovations are random with zero mean. This isn't the same thing because of Jensen inequality. Of course, if you consider only a linear approximation of the stochastic model, you are back with certainty equivalence.

That's it for the conceptual differences. Now for the Dynare implementation. In deterministic mode, simul, will compute a numerical simulation of the trajectory over the number of periods that you specify. Note that the algorithm makes the simplifying assumption that the system is back to equilibrium after the specified number of periods. For this reason you must specify a large enough number of periods such that increasing it further doesn't change the simulation for all practical purpose. This trajectory will basicaly describe how the system gets back to equilibrium after a unique shock in period 1.

In stochastic mode, stoch_simul will compute the first or second order approximated solution to your model and will optionnaly do a Monte Carlo simulation over the number of periods that you specify. In this Monte Carol simulation, you get a new random innovation is each period. The results will therefore be very different for the deterministic simulation. Now, with stoch_simul, you can also get IRFs, that is the expected future path of the variables conditional on a deterministic shock in period 1. In this case, the optional number of periods attached to the IRF option (IRF=n) specifies only the length of the IRF graph, without assuming anything about the return to equilibrium that remains asymptotic.

Because the certainty equivalence property of linear model, the IRF of a linearized version of the model computed with stoch_simul will be exactly the same as the simulation obtained with simul for the same linearized model, the same shock in period 1, and a simulation on a large number of periods to come as close as possible to asymptotic return to equilibrium.

In general, for nonlinear models, simul and stoch_simul will give you different results because simul takes into account the full nonlinearity of the model but ignore Jensen inequality and stoch_simul deals with Jensen inequality but computes only a first or second order approximation of the nonlinear model.

Kind regards

Michel