Steady State in Laubach/Williams Model

Steady state
1

Hi, I am currently trying to implement the natural rate estimation of Holston, Laubach and Williams (2003) in Dynare to get Bayesian Parameter estimates. My parametrization is pretty much according to my own ML estimates of the System.

I am not very experienced with Dynare, so I do not fully understand why the program gives me NaNs for my first two equations, so if someone could give me a hint what I should be looking for, that would be great. Thank you! My code is below. I could also provide the data if needed.

var y pi ypot g z str;
varexo e1 e2 e3 e4 e5 e6;
parameters root1 root2 a_r bpi by;

model (linear);
#ay1 = (root1+root2);
#ay2 = -root1root2;
y = ypot+ay1(y(-1)-ypot(-1))+ay2*(y(-2)-ypot(-2))+a_r/2*((-str(-1)-str(-2))+4g(-1)+4g(-2)+z(-1)+z(-2))+e1;
pi = bpipi(-1)+(1-bpi)(pi(-2)+pi(-3)+pi(-4))/3+by*(y(-1)-ypot(-1))+e2;
ypot = ypot(-1)+g(-1)+e3;
str = e6;
g = g(-1)+e4;
z = z(-1)+e5;
end;

varobs y pi str;

initval;
y= 1351;
pi= 2;
ypot= 1361;
str = -4;
g= 0;
z= 0;
e1= 0;
e2= 0;
e3= 0;
e4= 0;
e5= 0;
e6= 0;
end;

estimated_params;
root1, 0.95, -0.9999, 0.9999, uniform_pdf, 0, sqrt(3)^(-1)*1.9998;
root2, 0.55, -0.9999, 0.9999, uniform_pdf, 0, sqrt(3)^(-1)*1.9998;
a_r,gamma_pdf,-0.03,0.1,0;
bpi, gamma_pdf,0.4,0.2,0;
by, gamma_pdf, 0.05,0.2,0;
stderr e1, inv_gamma_pdf,0.01,inf;
stderr e2, inv_gamma_pdf,0.01,inf;
stderr e3, inv_gamma_pdf,0.01,inf;
stderr e4, inv_gamma_pdf,0.01,inf;
stderr e5, inv_gamma_pdf,0.01,inf;
end;

steady;
estimation(datafile=lw_obs,nobs=200,first_obs=5,mh_replic=2000,mh_nblocks=2,mh_drop=0.45,mh_jscale=0.8,mode_compute=1) ypot g z ;lw_rstar.mod (1.1 KB)

1 Like
2

Hi, I have just granted you permission to attach files in the posts. Please upload your mod file. I do not really know why but copy/paste does not behave well in posts (missing operators).

Best,
Stéphane.

3

Hi Stéphane, thanks! I uploaded the file.

4

Hi,

The steady state fails for two reasons:

  1. You did not give values to the parameters. The parameters need to be calibrated if you want to compute the steady state.

  2. Your model is linear and has unit roots. I do not understand from where you took the values in initval (these values are used as initial condition for the routine competing the steady state, clearly ypot=1361 and g=z=0 are not consistent) . In this case zero for all the variables is a more sensible choice (it is an obvious candidate for the steady state).

Note that because of the unit roots, you have an infinity of steady states for g, z and ypot (and the variables depending on them).

Because of the unit roots you will probably want to use a diffuse filter for the estimation.

Best,
Stéphane.

5

Thanks for your help!