# What is endogenous state variable in this simple model?

Dear DGSE experts,

The following model is famous simple 3 equation forward-looking NK model with forward-looking Taylor rule. I know the following model is easily solved using Dynare.

[code]model(linear);
//CB model
# kappa = (1-theta)(1-thetabeta)/theta;

``````	y	= y(+1) - (1/sigma)*(R - pi(+1)) + g;
pi     = beta*pi(+1) + kappa*y + z;
R      = rhoR*R(-1) + (1-rhoR)*(phipi*pi(+1)+phiy*y) + em;

// process of preference/cost shocks
g       = rhog*g(-1) + eg;
z       = rhoz*z(-1) + ez;
``````

end;[/code]

However, when I tried to solve this using Uhlig’s toolkit, I couldn’t figure out what is endogenous state variable in this model, which was necessary to construct matrices for Uhlig toolkit.

I know that capital is the endogenous state variable in RBC model, but there is no capital in this NK model. There are 3 endogenous variables (y, pi, R) in this model, and they don’t seem to be state variables.

Is there anyone who kindly help me to figure it out?

Regards,

Leo

Please someone help me to understand this…

The 3 equation NK model has no endogenous states. But depending on how you set up the code, it can be necessary to define the lags of endogenous variables as endogenous states, i.e. R(-1), g(-1), z(-1)

Thank you!

I think g(-1), z(-1) are rather exogenous state variables since g and z are exogenous processes? Thus, if we choose to make them state variables.

On R, I have seen a different specification in one of my macro class:

``````> R      = rhoR + (1-rhoR)*(phipi*pi(+1)+phiy*y) + em
``````

So that R becomes like an endogenous state variable. Thus, in period t, R(-1) is known. Actually in ‘Graduate Macro Theory II:A New Keynesian Model with Price Stickiness Eric Sims, 2015’, it mentions that R is endogenous state variable (page 33). So from your answer, it means they intentionally and purposefully made R endogenous state, not because it is, given that the 3 equation NK model has no endogenous states.

1. My reply referred to software implementations. There, usually all states are lumped together as being “endogenous”. Conceptually, lags of exogenous states are of course exogenous.
2. Your equation for R above is missing the lagged term. Again, the baseline NK model does not feature interest smoothing so there is no state variable. But if you add interest smoothing, a lagged interest rate shows up, which is a state.
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