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Re: st: Inequality constraints in a regression (new question)


From   Steven Samuels <sjsamuels@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Inequality constraints in a regression (new question)
Date   Sun, 14 Nov 2010 09:15:33 -0500

--

Herve-

This is exactly what I predicted in an earlier post, and is a consequence of the geometry of least squares. It occurs when the constraint is a4>=a2, but the original least square estimates have a2<a4. The LS solution with a4 = a2 has a lower residual sum of squares than any solution with a4>a2; that's what makes it "constrained least squares". You can get the constrained common coefficient directly by entering into -reg- the single covariate z =(g_ca + g_ca_t). Or use -nl- with the term {a2_4}*(g_ca + g_ca_t).

Steve

Steven J. Samuels
sjsamuels@gmail.com
18 Cantine's Island
Saugerties NY 12477
USA
Voice: 845-246-0774
Fax:    206-202-4783

On Nov 14, 2010, at 4:15 AM, Herve STOLOWY wrote:

Dear All:

Last week, I asked a question relating to inequality constraints in a
regression. I received several interesting answers who suggested to use
-nl- for the constraint: a2 < a4 and to replace a4 by  ({a2}+
exp({c}))*g_ca_t

Here is the new model.

nl (VTA = {a1}*g_t + {a2}*g_ca + {a3}*g_i + ({a2}+ exp({c}))*g_ca_t +
{a5}*g_ca_i + {a6}*g_t_i + {a7}*g_ca_t_i), initial(a1 1 a2 1 a3 1 c 1 a5
1 a6 1 a7 1)

To recover the value of a4, I use -nlcom-:

nlcom a4 : _b[a2:_cons] + exp(_b[c:_cons])

My problem is the following:

- in the first nl regression, I get a2 =  2.110647
- in the nlcom command, I get a4 = 2.110647. (I precise that all the
elements of the line seem to be identical: same standard error, same t,
same p, same confidence interval).

I am confused because I thought that this solution would make a4 higher
than a2, which is an important constraint to me. Here, a4 = a2.

Best regards

Hervé Stolowy


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Tel: +33 1 39 67 94 42 - Fax: +33 1 39 67 70 86
mail: stolowy at hec dot fr
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