Steady-State Parameter Relationships

Topics: Models
Apr 10, 2014 at 10:24 PM
Hi folks,

I have a new Keynesian model with a steady-state relationship of the same form as in the sticky price example - i.e. R = alpha * pi / beta. Because my model is linearized before being loaded into IRIS, R, alpha, beta, and pi are actually parameters in my case. pi, beta, and alpha are estimated and R is derived from the estimated values of those parameters in the !links section of the model file.

I am also modelling the yield curve in a Nelson-Siegel fashion. The measurement equations would be:

Image

Before including the yield curve model, the measurement equation for the short rate was:

Fed_Funds_rate = r + R; where R is a parameter that is derived from the relationship above.

I would like to replace the Fed_Funds_rate by a short maturity yield from the yield curve and would like to constrain the optimization procedure so that:
alpha * pi / beta = mu1 + 
          mu2  *  (1 - exp(-lamda * T)) / (lamda * T) + ...
          mu3  * ((1 - exp(-lamda * T)) / (lamda * T) - exp(-lamda * T);
where mu? are the mean reverting values of the factors L, S, and C and T is a short maturity yield.

What would be the most appropriate way of achieving this in IRIS?

Thanks!

Hector
Coordinator
Apr 11, 2014 at 7:10 AM
Hi Hector,

I think the equation R = alpha * pi / beta should actually be a dynamic relationship describing the interest rate reaction to inflation. This is your Taylor rule, no? I am working on something similar, so I can tell you how I think about it. What is typically done is to take some latent factor model (Diebold and Li (2006)):

Image

and augment it with some observable factors which are then linked to the latent factors as in Diebold, Rudebusch and Aruoba (2006):

Image

However, I don't see any reason that it's not possible to include a general equilibrium model with coefficient matrices A and B as:

Image

The question is then which macro states S to allow to affect the latent factors L. Presumably the policy rate is one of these, so this informs how you setup the coefficient matrix D.

I think what you're trying to do is restrict the dynamics of your short maturity (say, three month) treasury so that it follows a Taylor rule, which is sort of sensible. But what I think is better is to not include the fed funds rate in Y but rather put it in Z, let your general equilibrium model drive Z and then estimate the relationship to your yield curve, including the three month treasury.

But if you really want to create a non-linear constraint on parameters in an IRIS model estimation, then you have two choices:
  1. Read the section in help model/estimate on using a user-supplied minimization routine with extra input arguments. Use this to setup a call to fmincon in the optimization toolbox and include a non-linear constraint.
  2. Solve the constraint for one of the parameters and use this relationship to write one in terms of the other in the !links section. If you have a closed form expression for one parameter in terms of the others then this is obviously more efficient than (1) because you have fewer parameters and no non-linear constraint to deal with.
Mike
Marked as answer by jaromirbenes on 5/8/2014 at 9:23 AM
Coordinator
Apr 11, 2014 at 7:15 AM
You may also want to consider modifying your DSGE model so that it has a role through which long term interest rates may affect real activity. I think the most popular way of doing this currently is Andrés, López-Salido and Nelson (2004).
Apr 11, 2014 at 7:51 PM
Hi Mike, thank you for the response. The policy rule in the original DSGE model that I was working on was the traditional Taylor rule as you mentioned:
(r_t - R) = rho * (r_t-1 - R) + (1 - rho) * {beta1 * (ygap_t - YGAP) + beta2 * (pi_t - PI)} + e_t; where R, YGAP and PI are mean reverting levels.
My first approach to including the yield curve was very similar to the one suggested in your message. The reason of my initial inquiry is that when I generate forecasts using both the DSGE and the Diebold-Lee augmented approach the Fed Funds Rate evolved differently than the 3mo Treasury rate. Their mean reverting levels were also different. These two variables are almost identical and there should be a better approach of modelling the 'short' rate. My new Taylor rules would be similar to:
(l_t - L)  = rhoL * (l_t-1 - L) + (1 - rhoL) * (pi_t - PI) + el_t;
(s_t - S) = rhoS * (s_t-1 - S) + (1 - rhoS) * (ygap_t - YGAP)  + es_t;
           es_t = rhoSe * es_t + v_t
(c_t - C) = rhoC * (c_t-1 - C) + ec_t      
L, S and C conform the mean reverting level yield curve. I assumed that if I got rid of the Fed Funds Rate, as you are also doing in your work, but still respect the equilibrium condition from the DSGE would offer an appropriate solution to my problem. Hence the search of an alternative to enforce the following equilibrium relationship where T is 3mo:
alpha * pi / beta = L + 
          S *  (1 - exp(-lamda * T)) / (lamda * T) + ...
          C * ((1 - exp(-lamda * T)) / (lamda * T) - exp(-lamda * T);
Mike: I think what you're trying to do is restrict the dynamics of your short maturity (say, three month) treasury so that it follows a Taylor rule, which is sort of sensible. But what I think is better is to not include the fed funds rate in Y but rather put it in Z, let your general equilibrium model drive Z and then estimate the relationship to your yield curve, including the three month treasury.
My comment to your first sentence is Yes, that is what I'd like to achieve. I tried what you suggest in your second sentence and the reason why I'd like to avoid that is explained above. I guess there's not a easy fix....

Thanks,

Hector
Coordinator
Apr 14, 2014 at 7:38 AM
Hmmmm... interesting. Okay, I can see how it would be problematic to have notably different forecasts for the fed funds rate and three month treasuries. I think restricting parameters based on steady state relationships is unlikely to get you much in terms of what you want. But I might be wrong. Two other possibilities which come to mind are:
  1. Using system priors to place a preference on estimated models in which the fed funds rate and three month treasuries are highly correlated.
  2. Some modification of the model structure.
I will think about it. Sorry I can't offer better answers.
Marked as answer by jaromirbenes on 5/8/2014 at 9:24 AM
Apr 14, 2014 at 3:38 PM
Thanks for your response, Mike. I believe that your first alternative would still involve having both the Fed Fund Rate (FFR) and the Yield Curve (YC) in the model, and to model the YC in a Diebold-Rudebusch-Aruoba way. I will think about it.

In terms of the second alternative, my attempt to modify the model was that one explained in my previous reply. In this case, I would be replacing the traditional Taylor rule for 3 equations that would represent the new policy transmission mechanisms. I would also have a new transition equation/variable that would 'rebuild' the short rate (using T =3 for example) from the YC factors to replace the now gone FFR. The equilibrium value of the 'rebuilt' variable should ideally still respect the Keynesian model equilibrium value -- R = alpha * pi / beta.

This is fun. Thanks!

Hector