Trend growth of export in observation equation for a two-country DSGE Model

Dear Johannes and other scholars,
First thank you very much for your previous guidance, I am grateful.
I am currently formulating a two-country DSGE model and involves net exports appearing in observation equation, net exports=Country A’s export to country B-Country A’s import from country B*exchange rate,
I assume that country A has a trend growth rate g_A, country B has a trend growth rate g_B,
For country A’s output appearing in observation equation, I write:
For country B’s output appearing in observation equation, I write:
where obs_{yA} is observed output for country A, obs_{yB} is observed output for country B, yA_{t} and yB_{t} are log-differenced state variables for country A and B respectively

  1. Question 1, I have no idea about the trend for net exports, it should neither be g_{A} nor g_{B} right?
    Should the trend for net exports apearing in observation equation be g_{A}-g_{B}?
  2. Question 2, For echange rate entering as observable variable, should it be log-differenced or not?
  3. Question 3. exchange rate X has no trend in observation equation right?
    can the observation equation for exchange rate be obs_x=x_{t} or obs_x=x_{t}-x_{t-1}?
    where obs_x is observable variable for exchange rate, x_{t} is log-differenced state variable for exchange rate

Thank you very much and look forward to hearing from you.
PhD Candidate

This is tricky. There is the well-known issue that DSGE models predict a steady state for the net foreign asset position and therefore for net exports. That is typically not what we see in the data, potentially because the time series are too short. That makes it tricky to square the data that show trends in the trade balance with the model where such trends should not be present. I guess a common way to is the use the net export to GDP ratio as an observable and treat it as stationary.

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