Parameter constraints for MLE model

[Mark Liik]

Dear forum guests!

I need to introduce a range constraint for the sum on parameters in MLE model. This is necessary for economics reasons. At the moment, I have two ways to do that, inside of do-file. One is to add constraint directly to the sum of parameters, second one is to add constraint to every parameter separately. In last case, the sum is automatically constrained. I give you some example.

Suppose the parameters are b1, b2 and b3. Now, let's suppose that the sum is positive and less or equal to 1.5. Also, the sum is marked with the variable "scale". So, scale = 1.5*invlogit(b1+b2+b3), which means that 0≤scale≤1.5. In this case, no additional parameters.

The second option means I have to add some additional parameters, something like b01=invlogit(b1), b02=0.2*invlogit(b2) and b03=0.3*invlogit(b3). So, in that case scale=b01+b02+b03.
The choice of these multipliers is based on alternative estimates of parameters with another model. There is also some space left there.

Now becomes the tricky part. These two ways act very differently. The first option gives very realistic (economically) results. The sum of parameters is around 1.1. The second option gives not very realistic results, these parameters are not statistically significant, also likelihood has difficulties in converging.

But, I have been told, that this second option is correct and the first one is not? The reason, why this first option is not correct, is exactly because of the lack of these constraints directly for parameters. So, these constraints must be present in likelihood expression also, It is not enough that only the sum has constraints. In this case this is already some other model.
Is it really so?
Why should I use three constraints, when only one seems already enough? I would prefer to let the parameters vary freely and only restrict the range of their sum. Moreover, there is possibility to use "constraint" command separately, before the command "ml model". Only problem is that inequality constraint is not possible. But, in this case this is also only for the sum and is perfectly correct.
I am very grateful if anyone can explain and advise on this topic.

[FernandoRios]

Hi Mark
From what i gather from your explanations, the command constrain will not do what you need because, as you point out, it only covers cases of equalities not inequalities. In that scenario, the logit transformation or similar have always been my preference. In fact, that is what Stata does for the estimation of certain models, to restrict correlation parameters to be between -1 and 1, or variances to be strictly positive.
As you point out, both model specifications (restricting all parameters or each individually) are completely different. To say which one is the correct one for your purposes, depends on what is what you are trying to test. Do you want to test something regarding the overall "scale" variable, or you are interest in analyzing each b1 b2 and b3 parameter? answering this question will answer the question of which strategy you need to follow, the 1 restriction or 3 restrictions.
HTH

[Mark Liik]

Thank you for this explanation!
I am not trying to test particularly anything specific, I am just trying to estimate some parameter values, including these b1, b2 and b3. Based on the theory, these three parameters could be estimated also by the help of simple RE model.
But, in this case I can take those estimates only as indicative, because this simple RE model does not cover all related effects and include all other related explanatory variables I am looking for. I can't tell you, which one is more important, single parameter or their sum, they are related. But, it is much simpler to put constraint to the sum, instead of putting it individually to all parameters.
As I alreay mentioned, I am using these values of parameters as a reflection that the model is realistic. So, using only 1 constraint is simpler and gives me economically realistic results.But, is there any fundamental difference, which makes one option completely incorrect?