GPU (2010) replication: interest rate definition

Dear Prof Johannes,

I am trying to extend the model as in GPU (2010) and I found in you replication mod file equation 1, interest rate defined as:

r(t) = rstar + psi*(exp(d(t) - dbar) - 1) + exp(mu-1) - 1

I think in this formula it shoud be psi*(exp(d(t+1) - dbar) -1) since in the paper nowhere shows that r(t) related to D(t). r(t) is interest rate of debt issued at time t, and you defined “predetermined_variables k, d” so that it should appear d(t+1).

I think GPU made mistake here as all formula in the main text they wrote:

r(t) = rstar + psi*(exp(d(t+1) - dbar) - 1)

but in the appendix they nevertheless wrote:

r(t) = rstar + psi*(exp(d(t) - dbar) - 1)

It makes me confused or maybe I was wrong.

Could you please confirm which one is right?

Another question is how can you specify the observation equation as:

g_y = y/y(-1)*g(-1).

I do not understand this calculation though I read your guide. Basically, if Y = y*X(-1) then growth rate of Y is as

g_y = log(Y / Y(-1)) or g_y = log(y / y(-1)) + log(X(-1) / X(-2)) = log (y / y(-1)) + log(g(-1)).

In the model variable y is level so that g_y is gross growth rate. But the datafile you provide g_y, g_c and g_i are net growth rate. Could you please show me where I was wrong?

Thank you and best regards,

thanks for pointing this out. I have added a clarifying note to the file. I think that the equation in the paper is the one that makes sense (using d(+1), i.e. the debt level decided upon today). However, the results presented in the paper are generated using the specification in the Appendix with d, i.e. the debt level decided upon yesterday. This can be seen from the replication codes that also use this timing. I will notify the authors of this.

Regarding the observation equation: the equation is in levels, because the loglinear option will implicitly takes logs of all variables, making the equation log-linear. Normally, that option would trigger taking the log of the data as well. That is why the logdata option is specified to tell Dynare to not take the log again.

Thanh you very much Prof Johannes, I’ve got it.