When should intercept term be in the Taylor rule?

Dear Prof. Pfeifer, may I ask this question? In your codes for Gali’s book, Ch 3, you have the following;

[name='Interest Rate Rule eq. (26)']
i=phi_pi*pi+phi_y*yhat+nu;

And in the book, eq. 26 is i=\rho + \phi_{\pi} \pi_t+\phi_{y} \hat{y_t}+\nu_t. It seems whether to include intercept term in the rule depends on how you log-linearize the IS equation. Is that it?

A.
Gali linearizes the non-linear IS equation around a perfect foresight steady state with constant inflation π and constant growth γ, that is, i = ρ + π + σ γ which generates the log linearized IS equation, c_{t}=E_{t}\left(c_{t+1}\right)-\frac{1}{\sigma}\left(i_t-E_{t}\left(\pi_{t+1}\right)-\rho\right)+\frac{1}{\sigma}\left(1-\rho_Z\right) z_{t} which is consistent with i=\rho + \phi_{\pi} \pi_t+\phi_{y} \hat{y_t}+\nu_t.

B.
However, if we use the log-linearized trick, X_t=X e^{x_t}=X(1+x_t), then we get, c_{t}=E_{t}\left(c_{t+1}\right)-\frac{1}{\sigma}\left(i_t-E_{t}\left(\pi_{t+1}\right)\right)+\frac{1}{\sigma}\left(1-\rho_Z\right) z_{t} which is consistent with i=\phi_{\pi} \pi_t+\phi_{y} \hat{y_t}+\nu_t. Which, using Gali approach, would implying linearizing around a perfect foresight steady state with constant inflation π and constant growth γ, i = π + σ γ.

That is, it seems whether to specify an intercept in the rule depends on how you log-linearize the IS equation.




I found Gali explain it in another way. He says in the book, “Note that the choice of the intercept ρ makes the rule consistent with a zero inflation steady state.” In the original taylor rule, the intercept is \rho=i - \psi_{\pi} \pi, so if \pi=0, \rho=i=-\log \beta, which is equilibrium interest rate.

Now if we instead have i=\phi_{\pi} \pi_t+\phi_{y} \hat{y_t}+\nu_t, and log-linearize the IS equation with the trick, X_t=X e^{x_t}=X(1+x_t) and the assumption that R =\frac{1}{\beta} in steady state, what does it say about the intercept and the target inflation rate? The target inflation rate still zero, right? Although the intercept or \rho does not explicitly appear in the model equations.




And does it matter whether we use A or B?

It’s simply a matter of whether you leave the constant terms in the linearized equation or subtract them on both sides. Both options are mathematically equivalent, but you need to do this consistently.

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