In the An and Schorfheide model, is the interest rate a state or a static variable?

General DSGE Modeling
1

Hi all,

I’m trying to use this method https://www3.nd.edu/~esims1/linear_rational_expectations.pdf to solve the DSGE model of An and Schorfheide. Specifically, the linearized model is:

y_t = E_ty_{t+1} - \frac{1}{\tau}\big(R_t - E_t \pi_{t+1} - rho_z z_{t+1} \big) + (1-\rho_g)g_t \\ \pi_t = \beta E_t \pi_{t+1} + \kappa(y_t - g_t) \\ R_t = \rho_R R_{t-1} + (1-\rho_R)\psi_1\pi_t + (1-rho_R)\psi_2(y_t-g_t)+\epsilon_{R,t} \\ z_t = \rho_z z_{t-1} + \epsilon_{z,t} \\ g_t = \rho_g g_{t-1} + \epsilon_{g,t}

Rt is not a forward looking variable but it is affected by contemporaneous values of pi and y. The usual method to deal with such “static” variables that I’ve encountered is to just solve for them in terms of states and jump variables. However this does not seem possible because Rt is not just a linear combination of jump and state variables, but also contains lagged values of itself. It also doesn’t seem to be merely a state variable since it is affected by the jump variables.

I’ve been using this method described by Eric Sims which seemed close to the method that we discussed in our meeting on friday. I’ve gotten as far as finding the policy function for yt and pit in terms of the interest rate, zt, and gt, but cannot solve for Rt since it is also affected by y and pi.

I’m at a loss as to how to proceed from this point, and any help would be appreciated greatly.

2

The lag of R_t is a state variable in your model. This typically involves defining an auxiliary variable for the lag linking it to R_t.

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3

I suspected as much, thank you

4

If I have a backward looking inflation \pi_{t-1} in my NKPC, will this be interpreted as state now as explained in this one? The “Cogley, T. and Sbordone, A. M. (2008). Trend inflation, indexation, and inflation persistence in the new keynesian phillips curve. American Economic Review, 98(5):2101–26” has a partial indexation of inflation which makes the NK an AR(2) process.

5

Yes, everything dated t-1 is a state variable.

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6

To get more clarity under two scenarios:

  1. I have a model with habit formation (internal or external should not matter for this question). So c_{t-1} is state, c_{t} and c_{t+1} are control variables is it?

  2. I also have partial indexation in my model. So \pi_{t-1} is a state and \pi_{t} and \pi_{t+1} is a control?

Any references on these would help too. I have looked at the general taxonomy for state and control variables here - 26. Linear State Space Models — Quantitative Economics with Python.

7

Again, everything dated t-1 is a state variable. That applied to any variable appearing with a lag due to propagation mechanisms like indexing or habits.

8

Thanks. But my question is: can a state variable be also a control? Like c_{t-1} be state like you said, but c_{t} and c_{t+1} be control?

9

Again, to obtain the canonical AB-form, you need to introduce an auxiliary variable for c_{t-1}. This variable is a state, while c_t is the control.

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10

Thanks. I was trying the canonical AB version of the model. I will try this and post the codes.

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