2:21 p.m.
While preparing the notes for my econometrics class, I noticed that
for Table_10.5.gdt from the Gujarati dataset, Gujarati reports that
the Condition Index is found by SAS as \sqrt{3.0/0.00002422} (the
squareroot of the biggest eigenvalue divided by the smallest)
When I choose from the menu: View--Principal components, I see the
eigen values reported as 2.9720 and 0.0009 (instead of 0.00002422)
Can this be due to an accuracy error?
Regards
Talha
--
"Remember not only to say the right thing in the right place, but far
more difficult still, to leave unsaid the wrong thing at the tempting
moment." - Benjamin Franklin (1706-1790)
--
3:48 p.m.
On Mon, 18 Feb 2008, Talha Yalta wrote:
While preparing the notes for my econometrics class, I noticed that
for Table_10.5.gdt from the Gujarati dataset, Gujarati reports that
the Condition Index is found by SAS as \sqrt{3.0/0.00002422} (the
squareroot of the biggest eigenvalue divided by the smallest)
When I choose from the menu: View--Principal components, I see the
eigen values reported as 2.9720 and 0.0009 (instead of 0.00002422)
Can this be due to an accuracy error?
First, it seems you have run principal components on Y, X2 and X3, while
the condition number is computed on the X matrix (therefore, on a matrix
whose columns should be const, X2 and X3, suitably scaled).
Second, I don't know how SAS computes the condition number or if Gujarati
reports approximate figures, but there must be something wrong in
Gujarati's computations: if the eigenvalues refer to the X matrix with
columns scaled to have Euclidean norm 1, then the its cross-product matrix
must have ones on the diagonal and be positive (semi)definite. This means
that its trace is 3. Since all eigenvalues are non-negative and their sum
must be 3 (\sum_{i=1}^n \lambda_i = trace), it's impossible that the
largest eigenvalue is exactly 3 and the smallest one is positive.
Or maybe, the definition of condition number is something different. In
this case, I apologise.
<script>
open Table_10.5.gdt
matrix X = { const X2 X3 }
l = sqrt(diag(X'X))
matrix X = X ./ l'
XX = X'X
print XX
l = eigensym(XX, null)
print l
CN = maxc(l)/minc(l)
printf "CN = %g, (square root = %g)\n", CN, sqrt(CN)
</script>
Riccardo (Jack) Lucchetti
Dipartimento di Economia
Università Politecnica delle Marche
r.lucchetti(a)univpm.it
http://www.econ.univpm.it/lucchetti
6:47 p.m.
On Mon, 18 Feb 2008, Talha Yalta wrote:
While preparing the notes for my econometrics class, I noticed that
for Table_10.5.gdt from the Gujarati dataset, Gujarati reports that
the Condition Index is found by SAS as \sqrt{3.0/0.00002422} (the
squareroot of the biggest eigenvalue divided by the smallest)
When I choose from the menu: View--Principal components, I see the
eigen values reported as 2.9720 and 0.0009 (instead of 0.00002422)
Can this be due to an accuracy error?
Yes: the inaccuracy consists in using the wrong dataset, and not
taking logs ;-)
If you look carefully at Appendix 7A.5 you'll see that the SAS
output is based on data from Table_7.3.gdt, with logs taken of all
variables. Repeat what Jack suggested with the right data
open Table_7.3.gdt
logs Y X2 X3
ols l_Y 0 l_X2 l_X3
matrix X = { const l_X2 l_X3 }
l = sqrt(diag(X'X))
matrix X = X ./ l'
l = eigensym(X'X, null)
CN = maxc(l)/minc(l)
printf "CN = %g, (square root = %g)\n", CN, sqrt(CN)
and you'll find the same results as given by SAS (apart from the
slightly inaccurate reporting of the largest eigenvalue as 3.0 in
Gujarati, rather than 2.9996).
Allin.
7:29 a.m.
I have somehow missed that he was referring to Appendix 7.A.5. I
thought the example being mentioned was the one presented a couple of
pages earlier (Table 10.5).
Thanks very much for your help.
Cheers
Talha
On Feb 18, 2008 8:47 PM, Allin Cottrell <cottrell(a)wfu.edu> wrote:
On Mon, 18 Feb 2008, Talha Yalta wrote:
> While preparing the notes for my econometrics class, I noticed that
> for Table_10.5.gdt from the Gujarati dataset, Gujarati reports that
> the Condition Index is found by SAS as \sqrt{3.0/0.00002422} (the
> squareroot of the biggest eigenvalue divided by the smallest)
>
> When I choose from the menu: View--Principal components, I see the
> eigen values reported as 2.9720 and 0.0009 (instead of 0.00002422)
>
> Can this be due to an accuracy error?
Yes: the inaccuracy consists in using the wrong dataset, and not
taking logs ;-)
If you look carefully at Appendix 7A.5 you'll see that the SAS
output is based on data from Table_7.3.gdt, with logs taken of all
variables. Repeat what Jack suggested with the right data
open Table_7.3.gdt
logs Y X2 X3
ols l_Y 0 l_X2 l_X3
matrix X = { const l_X2 l_X3 }
l = sqrt(diag(X'X))
matrix X = X ./ l'
l = eigensym(X'X, null)
CN = maxc(l)/minc(l)
printf "CN = %g, (square root = %g)\n", CN, sqrt(CN)
and you'll find the same results as given by SAS (apart from the
slightly inaccurate reporting of the largest eigenvalue as 3.0 in
Gujarati, rather than 2.9996).
Allin.
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"Remember not only to say the right thing in the right place, but far
more difficult still, to leave unsaid the wrong thing at the tempting
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3 comments
3 participants
participants (3)
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Allin Cottrell -
Riccardo (Jack) Lucchetti -
Talha Yalta