Best Way to Find Stochastic Steady State

Stochastic simulations
1

Hi,

Maybe it has already been answered but I am wondering about the best way to find the stochastic steady state when using 3rd order approximation.

I usually simulate the model without shocks for 2000 periods and take the end point as the stochastic steady state.

What I tried last time was to further simulate the model with shocks for large number of periods and take the mean. I came to the realization that two ‘stochastic states’ can actually be different.

Note that I tried increasing the number of periods but I still get some difference which does not vanish away as I increase the number of periods.

I have come across a paper which proposes the following,

  1. Simulate the model without shocks 2000 periods, take the end point
  2. Simulate the model with shocks for 100 periods, take the mean
  3. Repeat (2) for 1000 times and take the mean of means

My prior would have been that all those methods should lead to reasonably close numbers but in practice my experience is that they can substantially differ. What do you suggest?

2

I have the impression that when people refers to the “Stochastic Steady State” they are usually referring to the method (1) in your list. However, is not expected that this procedure should deliver the true ergotic mean (retrieved from the procedures 2 and 3 ).

Indeed, since non-linear policy functions have cross products terms, the expected value of such terms is, in general, different from the end-point of a simulation with zero-valued shocks. This is for the same reason why E ( e_t^2) is a different number from (E (e_t) )^2.

I would rely on the procedures 2 and 3 to obtain long-run values for the non linear model.

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3

It depends on what you are trying to do. See e.g.
Risk matter comments EMAS steady state - #2 by jpfeifer and also Where is the stochastic steady state in dynare?

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