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Re: st: Fitting a linear regression where coefficients are bounded proportions

 From Martin Trombetta To statalist@hsphsun2.harvard.edu Subject Re: st: Fitting a linear regression where coefficients are bounded proportions Date Wed, 11 Dec 2013 17:27:14 -0200

Yes, it can be done, Nick. The problem is quite straightforward from a
mathematical point of view. The objective function is quadratic and
the constraints are linear.

Besides, think about it this way: imagine we only have two regressors.
Since parameters must be proportions, we might as well call them b and
1-b. All my inequality constraints say is that not all combinations of
b and 1-b are acceptable, only those that belong to a specific subset.
In that specific subset, there must be one combination that minimizes
the sum of square residuals and that is the one I want.

The problem is that I have 9 regressors instead of 2. I have
considered the possibility of using the multinomial logit approach,
but then I cannot guarantee that the inequality constraints will be
satisfied. Alternatively, I could use (appropriately defined)
individual logits for each coefficient, but then they would not sum to
one. I need an approach that guarantees both things.

2013/12/11 Nick Cox <njcoxstata@gmail.com>:
> Interval constraints are handled by reparameterisation; the logit is
>
> Constraints on parameter totals by one being total - sum of others.
>
> Your regression sounds so constrained that I wonder whether
>
> 1. You need data at all.
>
> 2. You have a real chance of getting it to fit.
>
> Nick
> njcoxstata@gmail.com
>
>
> On 11 December 2013 19:00, Martin Trombetta <martintrombetta@gmail.com> wrote:
>> Thanks Maarten, I had read this post before and, even though it was
>> useful at first, I think the methods suggested there do not quite help
>> with my particular problem. Please notice that I wish to include both
>> an equality constraint and several inequality constraints in the same
>> problem, I do not see how to use the methods from this post.
>>
>> 2013/12/11 Maarten Buis <maartenlbuis@gmail.com>:
>>> On Tue, Dec 10, 2013 at 7:50 PM, Martin Trombetta wrote:
>>>> I need to fit a linear regression where coefficients are to be
>>>> interpreted as proportions (that is, they must sum to 1), but at the
>>>> same time I wish to impose inequality constraints on each of them:
>>>> they all should belong to a specific interval (a,b) inside the unit
>>>> interval.
>>>
>>> There is a discussion of how to do that here:
>>> http://www.stata.com/support/faqs/statistics/linear-regression-with-interval-constraints/
>>>
>>> Hope this helps,
>>> Maarten
>>>
>>> ---------------------------------
>>> Maarten L. Buis
>>> WZB
>>> Reichpietschufer 50
>>> 10785 Berlin
>>> Germany
>>>
>>> http://www.maartenbuis.nl
>>> ---------------------------------
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>>
>>
>>
>> --
>> Martin Trombetta
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--
Martin Trombetta
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