# AR process for the foreign sector

Hi all,

I’m currently working in a non linear Small open economy model with a second order approximation and i have a question related to the foreign economy. I want to model the foreign sector as and AR process, but for the steady state computation of some of my home country variables, i need that the foreign interest rate and foreign inflation to have certain values, for example, i need the foreign interest rate to be approximately 4% (a gross steady state interest rate of 1.0095). My question is: is it correct to model the foreign interest rate as:

〖log⁡(R〗_t^)= λ^(R^ )+μ_t^(R^* )

then set the parameter λ^(R^* ) as λ^(R^* )=log⁡(1.0095) and the steady state value of the foreign interest rate as R^=exp⁡(λ^(R^ )) with μ_t^(R^* ) as a normal shock?

Can I do something similar for foreign inflation?

Best regards

Please make use of \LaTeX-capabilities to make the formula readable. Enclose the formula in .

I’m currently working in a non linear Small open economy model with a second order approximation and i have a question related to the foreign economy. I want to model the foreign sector as and AR process, but for the steady state computation of some of my home country variables, i need that the foreign interest rate and foreign inflation to have certain values, for example, i need the foreign interest rate to be approximately 4% (a gross steady state interest rate of 1.0095). My question is: is it correct to model the foreign interest rate as:

\log\left({{R^{*}}}_{t}\right)={\lambda^{*}}+{{\mu^{*}}}_{t}

then set the parameter λ^* as {\lambda^{*}}= \log\left(1.0095\right) and the steady state value of the foreign interest rate as {{R^{*}}}_{t}=exp({\lambda^{*}})
with {{\mu^{*}}}_{t} as a normal shock?

Can I do something similar for foreign inflation?

Best regards

Yes, typically you can do that, although AR-processes are more common. The only issue that may arise is that steady state interest rates faced by investors across countries may not be different as this would be inconsistent with a steady state.

So, in that case, I could use this autoregressive process, right?

log(R^*_t) = (1-\rho)log(R^*_{ss}) + \rho\log(R^*_t) + \mu_t

this also ensures a specific value of R^*_t in the steady state other than 1

Yes, except for the typo that it must be t-1 on the right.

Sorry for the typo and thank you professor.

Sorry professor, it’s me again, to specify foreign production as an AR process, I introduced the following equation to the model;

log(Y^*_t) = (1-\rho)*log( steady_ state (Y^*)) + \rho*log(Y^*_{t-1}) + \mu_t

The model runs without any problems, but, as expected, when I run model_diagnostics I get a singularity problem that arises from using the steady_state command. Given that the steady state of Y^*_t is different from one, and depends on other model variables and parameters, what is the most convenient way to avoid using the steady command ? Is it better to create model local variables?

When referring to the interest rate, using model local variables is not an issue because in the steady state it depends on parameters, but for Y^*, its value in the steady state is found recursively almost at the end, so I would practically have to repeat the steady state model in the form of model local variables.

Again, thank you very much
Eduardo

I am not sure I understand the problem. The above equation determines an exogenous process for Y^*_{t-1}. How can the steady state then be endogenously determined?

Thanks again for the reply. I am working in an open economy model, therefore foreign production enters only in the demand for exported goods X_t = (P^x_t/P^*_t)^{-\epsilon_f}*Y^*_t. For equilibrium in the balance of payments, I assumed that exports equal imports, therefore there is a specific value of Y^*_{ss} that guarantees the steady state of the balance of payments and the resource constraint. That value of Y^*_{ss} is the one I try to put in the AR process with the term log(steady_state (Y^*)). That is, there is a fixed value of Y^* that allows me to calculate the steady state of the model, but I want the evolution of Y^* around that fixed value to be an AR process.

It is something similar to what Adolfson et al (2007) do, they find the value of Y^*_t in the steady state, however, they model Y^*_t as a VAR (even though their model is linear). I don’t know if my approach is wrong, the external sector equations are the only thing holding me back from correctly closing the model.

Best regards

Can you provide me with the files? You may have one of the cases where you can ignore the singularity warning.

Sure professor, the mod file SOE_org is where I use the steady_state command, and where I get the singularity problem, in the mod file SOE, I solve the problem in a naive way, generating auxiliary parameters to define the steady state of Y^*_{ ss} recursively avoiding using the steady_state command, in which case model_diagnostics tells me that there is no problem, although obviously the results are exactly the same.

An easier way is to simply define one new parameter ystar_bar:
ystar_bar=ystar;