Interest rate rule about GK_2011

Hello everyone, I’ve been recently replicating the GK-2011 paper and encountered some confusion when dealing with nominal interest rates and interest rate rules.In his paper, the nominal interest rate is net,

i_t=(1-\rho)\left[\frac{1-\beta}{\beta}+\kappa_\pi \pi_t+\kappa_y\left(\log Y_t-\log Y_t^*\right)\right]+\rho i_{t-1}+\varepsilon_t\\ 1+i_t=R_{t+1} \frac{E_t P_{t+1}}{P_t}

If we define $i_t$​ is gross nominal interest rate , then

i_t=(1-\rho)\left[\frac{1}{\beta}+\kappa_\pi \pi_t+\kappa_y\left(\log Y_t-\log Y_t^*\right)\right]+\rho i_{t-1}+\varepsilon_t\\ i_t=R_{t+1} \frac{E_t P_{t+1}}{P_t}

If we further define i_t as log gross nominal interest rate, then

e^{i_t}=(1-\rho)\left[\frac{1}{\beta}+\kappa_\pi e^{\pi_t}+\kappa_y\left( Y_t- Y_t^*\right)\right]+\rho e^{i_{t-1}}+\varepsilon_t\\ e^{i_t}=e^{R_{t+1}} e^{\pi_{t+1}}

However, it is not same with the interest rule in his FA.mod file. Actually, I think in his mod file the definition of i_t is different between these equations, where in interest rule i_t is gross nominal interest rate and in fisher equation it is log gross nominal interest rate.

Thanks in advance!

In his code, he used gross nominal rate and a non-linear Fisher equation with gross rates. So no problem at all. Have a look at lines 95 and 98.
FA_model.m (3.9 KB)

I guess the confusion comes from log-convention. So basically eq (1) is i = \text{log}\; (\frac{1}{\beta})

If the issue is not already settled, can you please clarify which equations and/or codes you are referring to,

Let me describe my questions thoroughly. The first confusion arises from the definition of gross nominal interest rate. In Gali(2015), the interest rate is defined as

i_t = \ln R_t = -\ln Q_t

and the interest rate rule is

e^{i_t}=\frac{1}{\beta}\left(\Pi_t\right)^{\varphi_\pi}\left(\frac{Y_t}{Y}\right)^{\varphi_y} e^{v_t}

The corresponding dynare codes are

i = log(R)

exp(i) = (1/beta)*(pi)^(phi_pi)*(Y/steadystate(Y))^(phi_pi)*e^(v_t)

If we focus on the percentage deviation from the steady state, the codes can be modified as

exp(i) = R

The confusion lies in the second equation, is it

exp(exp(i)) = (1/beta)*(exp(pi))^(phi_pi)*(exp(Y)/steadystate(exp(Y)))^(phi_pi)*e^(v_t)?

Back to GK, here the definition of gross nominal interest rate is


The interest rate rule is

i_t=(1-\rho)\left[1/\beta+\kappa_\pi \pi_t+\kappa_y\left(\log Y_t-\log Y_t^*\right)\right]+\rho i_{t-1}+\varepsilon_t

If we now focus on level deviation, then dynare codes are

i = R*pi(+1)

exp(i) = exp(i(-1))^rho_i*((1/beta)*exp(pi)^kappa_pi*(exp(X)/(epsilon/(epsilon-1)))^(kappa_y))^(1-rho_i)*exp(e_i);

Now if we focus on percentage deviation, the fisher equation should be

exp(i) = exp(R)*exp(pi(+1))

What about interest rate? Why in his code FA.mod (7.4 KB), the interest rate is still

exp(i) = exp(i(-1))^rho_i*((1/beta)*exp(pi)^kappa_pi*(exp(X)/(epsilon/(epsilon-1)))^(kappa_y))^(1-rho_i)*exp(e_i);?

The only difference between the two papers is the fact that Gali uses the exact definition of i_t=\ln R_t while GK make use of an approximation \ln 1+x_t\approx x_t for small x_t, so that for 1+i_t=R_t you get that i_t\approx \ln R_t.