Re: FW: st: cnsreg with singular

[Tirthankar Chakravarty <tirthankar.chakravarty@gmail.com>]

Demetris,

The answer to your question involves considerable algebra as well as
knowing what Stata does with the constraints you supply to it. I have
done the algebra for you in a document I have uploaded here:
http://pdfcast.org/download/statalist-query.pdf

Basically, I have shown you the algebraic (closed-form)
quasi-equivalence between the two solutions and a Stata example to
illustrate this. No iterative optimization algorithms are required or
are used by Stata.

You will also need to look at the manual entry in [P] for -makecns- to
see further algebraic manipulations - if I have time, I will add these
also to the document also.

T

PS> Note the typo in the last line of eqn. 11; a 1/3 is missing.

On Wed, Sep 7, 2011 at 6:44 AM, Cameron McIntosh <cnm100@hotmail.com> wrote:
> No problem, hope you find the references helpful... but sorry, I don't know what cnsreg does behind the scenes in such a case. So the various manual treatments of the problem may or may not be better, I'm not sure. :)
> Cam
>
>> Subject: Re: st: cnsreg with singular
>> From: Demetris.Christodoulou@sydney.edu.au
>> Date: Wed, 7 Sep 2011 21:24:27 +1000
>> To: statalist@hsphsun2.harvard.edu
>>
>> Thanks for the very useful references Cam, these will keep e busy for a while!
>>
>> Still, can someone please describe the current mechanics of cnsreg in the case of a singular design matrix?
>>
>> many thanks, Demetris
>>
>> On 07/09/2011, at 10:57 AM, Cameron McIntosh wrote:
>>
>> > Hi Demetris,
>> >
>> > I wonder if it would also be worthwhile to try some corrective procedures on the design matrix, and see how these compare to the built-in methods in cnsreg?
>> > Yuan, K.-H., & Chan, W. (2008). Structural equation modeling with near singular covariance matrices. Computational Statistics & Data Analysis, 52(10), 4842-4858.
>> >
>> > Yuan, K.H., Wu, R., & Bentler, P.M. (2010). Ridge structural equation modelling with correlation matrices for ordinal and continuous data. British Journal of Mathematical and Statistical Psychology, 64(1), 107–133.
>> >
>> > Bentler, P.M., & Yuan, K.-H. (2010). Positive Definiteness via Offdiagonal Scaling of a Symmetric Indefinite Matrix. Psychometrika, 76(1), 119-123. http://www.springerlink.com/content/k5154122171551l2/fulltext.pdf
>> >
>> > Highham, N.J. (2002). Computing the nearest correlation matrix - a problem from finance. IMA Journal of Numerical Analysis, 22(3), 329–343.
>> >
>> > Knol, D.L., & ten Berge, J.M.F. (1989). Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika, 54, 53–61.
>> >
>> > Are you using the model option "col" (keep collinear variables)? Sorry if I am off base given the substantive and methodological nature of your analysis (which I don't know).
>> >
>> > Best,
>> >
>> > Cam
>> >
>> >> From: demetris.christodoulou@sydney.edu.au
>> >> To: statalist@hsphsun2.harvard.edu
>> >> Date: Wed, 7 Sep 2011 09:50:35 +1000
>> >> Subject: st: cnsreg with singular
>> >>
>> >> My question is how does cnsreg deals with a singular matrix?
>> >>
>> >> Consider the following example:
>> >>
>> >> . sysuse auto
>> >> . generate mpgrep78 = mpg + rep78
>> >> . regress price mpg rep78 mpgrep78
>> >>
>> >> Due to perfect collinearity (i.e. a singular design matrix), linear OLS drops one of the explanatory variables.
>> >> But I can force 'estimation' by:
>> >>
>> >> . constraint 1 mpgrep78 = mpg + rep78
>> >> . cnsreg price mpg rep78 mpgrep78, cons(1)
>> >>
>> >> This produces estimates for all three explanatory variables.
>> >> I noticed that the estimates of cnsreg are exactly the same, as taking the estimates of regress and apply the linear relationship to calculate the third parameter.
>> >>
>> >> This is what Greene (2010, p.274) suggests as well but in a more elaborate context using multiple regressions. That is, estimate the M-1 parameters and then use the linear relationship to calculate the M parameter.
>> >> Can someone please confirm whether this is what Stata does too?
>> >>
>> >> Or does it use some more complex iterative numerical optimisation procedure, perhaps even involving a singular value decomposition?
>> >>
>> >> I am using Stata/MP2 version 11.2 on Mac.
>> >>
>> >> many thanks in advance,
>> >> Demetris
>> >> *
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>> >
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>>
>>
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>
> *
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>



-- 
Tirthankar Chakravarty
tchakravarty@ucsd.edu
tirthankar.chakravarty@gmail.com

*
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