# RE: st: Re: Finding a vector minimizing the variance with Stata 8.2

 From "Feiveson, Alan H. (JSC-SK311)" To Subject RE: st: Re: Finding a vector minimizing the variance with Stata 8.2 Date Mon, 17 Jul 2006 09:32:30 -0500

```Kristin - If you min w'Vw subject to w'e=1 where e is a vector of ones,
the solution is w = VIe/(e'VIe) where "VI" stands for V-inverse. You can
derive this by using a LaGrange multiplier - L
min w'Vw - 2L(w`e) => Vw-Le = 0 => w = LVIw. But since e'w=1, L must be
equal to 1/e'VIe.

However this does not guarantee that the elements of w will be
non-negative. If you require w>=0 then you have a quadratic programming
problem. If the unconstrained solution has one or more negative
elements, then it can be shown that the constrianed solution is
equivalent to solving the unconstrained problem with some of the
elements equal to zero. You could try all the combinations to see which
gives you the minimum.

Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Kristin J.
Kleinjans
Sent: Monday, July 17, 2006 8:58 AM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Re: Finding a vector minimizing the variance with Stata
8.2

Hi Michael,
that seems obvious - sorry, I should have explained what I am trying to
do ...

I am trying to find a linear combination of random variables a1-a4 which
minimizes the variance. So I am trying to find weights w=(w1 w2 w3 w4)
such that a=w1*a1+w2*a2+w3*a3+w4*a4  subject to  w1+w2+w3+w4=1 has the
lowest possible variance.

(I am estimating a structural model with overidentifiying restrictions
for the parameters I am interested in. I have 4 nonlinear combinations
of estimates for the parameter of interest. I get those using the nlcom
command.)

Hope this explains my problem better!
Thanks,
Kristin

> How about w=0?    ;)
>
> M Blasnik
>
> ----- Original Message -----
> From: "Kristin J. Kleinjans" <kkleinjans@econ.dk>
> To: <statalist@hsphsun2.harvard.edu>
> Sent: Monday, July 17, 2006 9:27 AM
> Subject: st: Finding a vector minimizing the variance with Stata 8.2
>
>
>> Dear Statalist,
>> I need to find a vector w such that
>> X=w'V w is minimized, where V is a variance-covariance matrix. I use
>> Stata 8.2 - does anybody know if there is a way to do this?
>>
>> Thank you,
>> Kristin
>
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```