# Steady state of a random process

Hello,

in my model I have the following process for technology:

s_{t}=exp(x_{t}-(sigma^2/2(1-rho^2)))
x_{t}=rho*x_{t-1}+epsilon_{t}
where epsilon_{t} is normal and has mean 0 and variance sigma^2.

Now, the expected value of s_{t} is 1 and its steady state value should be 1 as well (given that x_{0}=0).

The problem is that if input this process in dynare in the following way:

var s,x;
varexo epsilon;
parameters sigma, rho;

shocks;
var eps=sigma^2
end;

model;
s=exp(x_{t}-(sigma^2/2(1-rho^2)))
x=rho*x(-1)+epsilon
end;

stoch_simul;

In the results I get that the first moment of s is 1, but its steady state value is not 1, but equal to exp(-sigma^2/2(1-rho^2)).
How can I avoid this problem? Is there a way to tell dynare that the parameter “sigma” must be equal to zero when computing the steady state (and only when computing the steady state)?

Fabrizio

Sorry, but I don’t get your question. A parameter is a parameter, i.e. it is fixed and time-invariant and never changes between steady state and the neighbourhood of the steady state. If the code you posted should account for Jensen’s Inequality terms occuring at higher order perturbations, it should result in the policy functions from the approximation of the nonlinear equations and should not be entered manually in the way you do.

Sorry if I wasn’t clear enough in my first post.

What I’m trying to do is to input the following process in Dynare:
http://img855.imageshack.us/img855/6474/processi.png
Such that the expected value of s_{t} and its steady state value are equal to one.

My first attempt was to write down “sigma^2” as a parameter, but in that case I got that

E_{t}(s_{t})=1

while

http://img38.imageshack.us/img38/2597/19971617.png

Is there a way to make dynare understand that the “sigma^2” in the expression of s_{t} is the variance of epsilon and, thus, it should be equal to zero in the steady state?

Dear Fabrizio,

what you want to do is not currently possible in Dynare. To do what you want, I think that sigma should be the perturbation parameter that we expand along to go from the deterministic model towards the stochastic one. Currently there is no way to refer to this parameter in the model and we should add this feature. It is on my todo list …

Best

Michel

Dear Michel,

thank you for your answer and for your work with Dynare. I’ll wait for the next version, in the meanwhile I’ll use a different process.

Best,

Fabrizio