# Mean and steady state values in a Small Open Economy

Hi everyone,

I’m currently working on a small open economy (SOE) model. For simplicity, I have assumed that inflation in the steady state of the small economy is equal to inflation in the rest of the world (\Pi = \Pi^*), as well as the interest rate (R = R^*). The data I’m using for the “rest of the world” is based on the United States. However, the issue arises as the interest rate in the US is, on average, lower than in my small economy data (Mexico), and the same applies to inflation. Consequently, the mean of the data for the US variables will differ from their steady-state values in the model (the foreign sector or rest of the world variables are modeled as an AR(1) process). I’m not sure what to do:

1. Let the shocks account for the difference between the steady-state values and the mean of the data for the US (I’m not sure if this is actually correct since, in the real world, one can assume that in steady state R and \Pi would differ across countries).

2. Add a constant term in the observable equations of the US variables to match the steady-state value with the mean of the data.

3. Explicitly account for different steady-state values R \neq R^* and \Pi \neq \Pi^* in the model. Although this creates huge problems to solve the steady state. If there is any literature explicitly modeling differentials in interest rates and inflation in a small open economy setting, I would appreciate it if you could share it with me.

I opt for the first option, but I’m not sure if it’s correct.

Thanks in Advance.

Can you outline how how the foreign variables enter? Is the model nonlinear or do we only care about deviations from steady state? Ultimately, in most models it only matter what the steady state real interest is, not the individual components. Do you know whether the implied real interest rates in your two regions are the same?

Thank you for your response.

The model is in non-linear form. The foreign sector (corresponding to the US data) is modeled as an AR(1) process. The interest rate of the rest of the world enters in the UIP condition only, and \Pi^* enters in the definition of relative prices and the marginal cost of a tradable sector of the small open economy.

Indeed, even real rates have historically differed, with higher rates in Mexico. If I take the last 20 to 25 years, the interest rates have differed by about 100 to 150 bps, less than the nominal rates (500 bps), but ultimately they differ.

Would it be correct to let the shocks explain the difference between the data means and steady state values for the US?

I have estimated the parameters using the first two approaches outlined above, but they yield very different results.

Thanks again for your help.
Greetings

No, in order for shocks to account for large difference in the mean, your estimated model needs to have a unit root. Is there a reason you cannot simply work with deviations from the their respective means? That would eliminate the differential trends.

Sorry, but what do you mean by this?

Do you mean deviations from their means in the observable equations, or in the model in general?

Thank you for your help.

I mean for the VAR part of the exogenous processes. Simply impose the steady state of the foreign variables to be identical to domestic ones and then use a VAR in deviations from the actual data mean. That makes the data and model trends compatible.