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In order to replicate recent NK models which implement the Zero Lower Bound, so an interest rate which is subject to the constrained of being positive,
I have derived a continuous version of the Kronecker Delta function(an IF function) for the monetary policy rule, so it gives Rzlb(R)=R for R>0 and Rzlb(R)=0 for R<0.

The function I used is:
Rzlb = k*log(1-exp(-1/k)+exp(r/k));

Which perfectly approximates the "If" function for when k tends to zero.
So I have set k=1/1000 #so some very small value ( a plot can be seen here: http://www.wolframalpha.com/input/?i=%281%2F1000%29*log%281-exp%28-1%2F%281%2F1000%29%29%2Bexp%28x%2F%281%2F1000%29%29%29 )

Also the taylor series approximation follows the "If" function, although it is dependant from which value the approximation is made whether it will give zero everywhere or Rzlb=R everywhere.
I figured for small deviations in R this shouldn't be a problem, only when crossing between negative and positive values.
however, after I ran some models, and saved the output using the dynasave command, I found when comparing the Rzlb output with its R input, the Rzlb function still gave negative values.

I am using the Stoch simulate command, and the function within the log shouldnt be zero.

Any ideas why it doesn't work?

Code: Select all

y = y(+1) - 1/sigmaC*(r-pi(+1)) + s_b;
   pi = beta*pi(+1) + ((1-theta)*(1-beta*theta)/theta)*(sigmaC+sigmaL)*y + s_p;
   % Monetary Policy Rule
   
   % R input where rz is the unbounded r, and r is the interest rate constrained to be positive
   rz = rho*r(-1) +  (1-rho)*( phi_pi*pi + phi_y*y ) + phi_dy*(y-y(-1)) + s_r  ;
    r = k*log(1-exp(-1/k)+exp(rz/k));

If I understand you correctly, you are basically using a barrier function approach. That does not get you around the problem that a first order Taylor approximation is going to replace your nonlinear function by a linear one. An linear functions by construction are unbounded.

Thank you for the answer!