## How to restrict parameters of smooth transition models

Discussion of models with structural breaks or endogenous switching.

### How to restrict parameters of smooth transition models

Dear all,
I am trying to estimate a logistic smooth transition model. Is there any way allowing me to restrict the nlls estimate of the threshold value c within a specific range (e.g. [1.6, 1.7] )?
My transition variable has a range of [1.0, 2.2]. Right now I just restrict the initial value of c within [1.6, 1.7] and use a grid search over c to determine the initial values giving maximum log likelihood, but I find the final nlls estimate would likely give a value falling outside the range (e.g. 1.3), which does not make sense from the data visualization.
Is the big gap as a result of a convergence issue? Do you have some ideas about restricting a range for NLLS estimates? Thanks for your time and consideration.

Below is my code.

Code: Select all
`* set the list of free parametersnonlin(parmset=gparm) eta c* create transition functionsstats(noprint) scom sigma=sqrt(%variance)frml gf = %logistic(exp(eta)*(s-c)/sigma,1.0)linreg(noprint) y# constant xcom reglist=%reglist()equation eq1 y# reglistequation eq2 y# reglist* define frml for nonlinear estimationfrml(equation=eq1,vector=parm1,parmset=set1) parm1ffrml(equation=eq2,vector=parm2,parmset=set2) parm2ffrml stc y = g=gf,parm1f+g*parm2fcom regparm=set1+set2com ll_best=-100.0@gridseries(from=1.6,to=3.5,size=0.1) grid_etacom ngrid=20@gridseries(from=1.6,to=1.7,n=ngrid) grid_c* search initial values giving max logldo i_eta=1,20      do i_c=1,ngrid         com eta=grid_eta(i_eta), c=grid_c(i_c)         sweep(group=(s>c),var=hetero,ibeta=ibeta)         # y         # reglist         com parm1=ibeta(1),parm2=ibeta(2)-ibeta(1)         nlls(noprint,parmset=regparm,frml=stc) y         if %logl>ll_best {            com ll_best=%logl,\$            eta_best=eta,c_best=c,\$            parm1_best=parm1,parm2_best=parm2            }      end do i_cend do i_eta* set initial valuescom eta=eta_best,c=c_best,parm1=parm1_best,parm2=parm2_best* NLLS estimationnlls(parmset=regparm+gparm,frml=stc) y`
wenbei

Posts: 8
Joined: Fri May 29, 2020 1:54 pm

### Re: How to restrict parameters of smooth transition models

You can use the constrained optimization on NLLS---though that's probably overkill in this case since the global (constrained) optimum will be at 1.6. (It has to be at one of the end points if the unconstrained optimum isn't inside the interval.) I'm not sure what you mean by "data visualization" though? This is a break in the linear relationship between y and x using values of s.
TomDoan

Posts: 7147
Joined: Wed Nov 01, 2006 5:36 pm

### Re: How to restrict parameters of smooth transition models

Thanks for your timely reply. This sounds like exactly what I need. The transition variable s I am using is actually the variable x. I scatter y versus x (vertical versus horizontal axis) and find there could be a natural break at 1.6 in their relationships. But the nlls gives me 1.3, in which case the result visualization (fitted y versus x) look a bit weird. Anyway, thanks a lot. I’ll give a try on the constrained optimization and see what happens.
wenbei

Posts: 8
Joined: Fri May 29, 2020 1:54 pm

### Re: How to restrict parameters of smooth transition models

TomDoan wrote:though that's probably overkill in this case since the global (constrained) optimum will be at 1.6. (It has to be at one of the end points if the unconstrained optimum isn't inside the interval.)

Hi Tom,
Why does it have to be at one of the end points? This sounds like the optimization of a monotonous function has corner solutions. I thought it would be often the case that the function of fitting criteria is nonlinear and it achieves local optimum at a point within the constrained range. Do I miss anything? Thanks for your consideration.
wenbei

Posts: 8
Joined: Fri May 29, 2020 1:54 pm

### Re: How to restrict parameters of smooth transition models

Isn't this (from the original reply) pretty much what you're saying: "It has to be at one of the end points if the unconstrained optimum isn't inside the interval."
TomDoan

Posts: 7147
Joined: Wed Nov 01, 2006 5:36 pm

### Re: How to restrict parameters of smooth transition models

Thanks. Take my dataset as an example. I find the unconstrained (global) optimum gives an estimate of 1.4, and then I would like to constrain it within [1.6, 1.7]. You told me the constrained optimum has to be at one of the end points if ... and I do find that it is at 1.6. But I was also wondering if this result would be certain or coincident. Could it has constrained (local) optimum within [1.6, 1.7] and the estimate would be at some value like 1.63, 1.66 or 1.68?
wenbei

Posts: 8
Joined: Fri May 29, 2020 1:54 pm

### Re: How to restrict parameters of smooth transition models

If you have multiple local modes, nothing is really certain. That's where the annealing algorithms can be useful as they search more broadly. However, if you do a grid search restricted to your constrained interval, and it doesn't find any sign of a local mode on the interior, then you can probably be pretty safe in assuming that there isn't one.
TomDoan

Posts: 7147
Joined: Wed Nov 01, 2006 5:36 pm

### Re: How to restrict parameters of smooth transition models

I see. Much appreciated for your patient explanations.
wenbei

Posts: 8
Joined: Fri May 29, 2020 1:54 pm