Hi everyone,

If I have a non-linear DSGE model and I’ve decided that the external sector (\Pi^*, Y^*, R^*) would follow an AR(1) process, what is the correct way to specify these equations?

log(R^*_t) = (1-\rho_{R^*}) \cdot \log(R) + \rho_{R^*} \cdot \log(R^*_{t-1}) + \mu_{R^*t}

or

R^*_t = (1-\rho_{R^*}) \cdot R^* + \rho_{R^*} \cdot R^*_{t-1} + \mu_{R^{*}t}

The same thing with Gross inflation:

\log(\Pi^*_t) = (1-\rho_{R^*}) \cdot \log(\Pi) + \rho_{\Pi^*} \cdot \log(\Pi^*_{t-1}) + \mu_{\Pi^*t}

or

\Pi^*_t = (1-\rho_{R^*}) \cdot \Pi + \rho_{\Pi^*} \cdot \Pi^*_{t-1} + \mu_{\Pi^*t}

Both methods provide fairly close estimates, but ultimately, they differ.

Similarly, if I were to model the foreign sector as a VAR, should I estimate the VAR with the natural logarithm of gross inflation and gross interest rate, or should I estimate the VAR with the levels of these variables? It makes more sense to apply the natural logarithm to the foreign output gap, but regarding the interest rate and inflation, I’m not entirely clear.

Thank you.