# Foreign sector in a estimated non-linear model

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.

At first order, both specifications should be equivalent as long as you correctly scale the standard deviation of the shock.

Thank you for your response, Professor.

Building upon these ideas, if I opt to incorporate the foreign sector into the nonlinear model as an SVAR, I’m somewhat confused with the Dynare representation. I require a second-order approximation to do a welfare analysis. So, imagine that my SVAR (with variables in log deviation from steady state) in matrix form is:

\hat{X}_t = A_0^{-1}F(L) \hat{X}_{t-1} + A_0^{-1}D\mu_t

Where A_0 is the contemporaneous effect matrix of the form:
A_0=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ -\delta_{3,1} & -\delta_{3,2} & 1 \\ \end{bmatrix}

while \mu_t denotes the structural shock, following \mu \sim N(0, \Sigma_{\mu}) and D constitutes a diagonal matrix containing standard errors along the diagonal. As I’ve estimated the SVAR outside of Dynare, I’m somewhat confuse about how to incorporate it into Dynare. Would it be correct if, in the model block, I specify the following equations?:

\hat{\Pi}^*_{t} = \alpha_{1,1} \hat{\Pi}^*_{t-1} + + \beta_{1,1} \hat{Y}^*_{t-1} + \gamma_{1,1} \hat{R}^*_{t-1} + \mu_{\Pi,t}

\hat{Y}^*_{t} = \alpha_{2,1} \hat{\Pi}^*_{t-1} +\beta_{2,1} \hat{Y}^*_{t-1} + \gamma_{2,1} \hat{R}^*_{t-1} + \mu_{Y,t}

\hat{R}^*_{t} = \alpha_{3,1} \hat{\Pi}^*_{t-1} + \beta_{3,1} \hat{Y}^*_{t-1} + \gamma_{3,1} \hat{R}^*_{t-1} + \delta_{3,1}*\mu_{\Pi,t}+ \delta_{3,2} \mu_{Y,t} + \mu_{R,t}

And then, in the shocks block, should I specify the standard errors of the shocks I estimated outside Dynare? I mean, should I write the elements of D in the Shocks block?, for instance:

var \mu_{\Pi,t};
stderr 0.001582;