In article <[EMAIL PROTECTED]>,
Richard A. Beldin, Ph.D. <[EMAIL PROTECTED]> wrote:
>I suggest that you need to have some criterion to use as a tie breaker.
>Can you identify any of the observed characteristics which might be a
>proxy for a "goodness-of-fit" criterion?
Thanks for posting the equations in a form which I can
read without help from our site manager. This is, for
me, very old stuff, IF my interpretation of the assumptions
is correct. If it is not, they need to be clarified.
One must keep in mind that the consultant is not the one
to make the assumptions.
>Kenneth Benoit wrote:
>> Dear Colleagues:
>> Perhaps someone can help me with this problem. I am trying to solve for
>> a number of parameters in three equations which are linked through
>> composition of the data. each model yields different parameter
>> estimates when estimated alone since the parameters are overidentified.
>> I'd be happy for any advice on the problem!
>> Apologies for cross-postings.
>> Ken Benoit
>> ---------------------------------------------------------
>> Kenneth Benoit http://benoit.tcd.ie
>> Department of Political Science mailto:[EMAIL PROTECTED]
>> Trinity College Tel: 353-1-608-2491
>> Dublin 2, Ireland Fax: 353-1-677-0546
>> Consider a system where:
> Y1 = X0 + (1-g11)b1 X1 + g21b2X2 + g31b3X3
> Y2 = X4 + g12b1 X1 + (1-g22)b2X2 + g32b3X3
> Y3 = X5 + g13b1 X1 + g23b2X2 + (1-g33)b3X3
>> and:
>> 1 = Y1 + Y2 + Y3, 0 Yi 1.0 "i
>> 1 = X0 + X4 + X5, 0 Xi 1.0 "i
>> 1 = g11 + g12 + g13, 0 gij 1.0 "i,j
>> 1 = g21 + g22 + g33
>> 1 = g31 + g32 + g33
>> GOAL: To estimate g's and b's. Problems: overidentification; effects
>> of the data items and some of the parameters summing to 1 which I
>> still don't fully understand.
>> Background: This is for a voting transition study in Italy, where the
>> b's represent a the proportion of voters following a rational
>> proximity model, and the g's represent the discrete probability
>> distribution according to which non-rational voters distribute their
>> votes to one of three electoral coalitions (corresponding to the Y's).
>> I have data for all of the Y's and X's, which are proportions.
>> Possible ways to simplify:
>> * Set gij's to constants before estimation.
>> * Set gij = g* " i,j.
>> * Set b3 = 1.
>> * Set b1 = b2.
It is not necessary to do anything like this. The problem
is easy enough, using old methods.
>> File translated fromTEXby TTH,version 2.56.
>> On 26 Nov 1999, 14:00.
What I believe is Ken's model (I have only the printed version
to go by) is
Y1 = X0 + (1-g11)b1 X1 + g21b2X2 + g31b3X3 + U1
Y2 = X4 + g12b1 X1 + (1-g22)b2X2 + g32b3X3 + U2
Y3 = X5 + g13b1 X1 + g23b2X2 + (1-g33)b3X3 + U3,
the U's being the disturbances in the equations. From the
stated information, the U's add up to 0 as well; there are
really only two equations. But this means that the U's
cannot be independent. Also, it seems that the terms
1-gii should be gii-1; this is needed for the equations
to add up. I may be wrong, but I will assume this correction
to be made.
Now set D1 = Y1 - X0, D2 = Y2 - X4, D3 = Y3 - X5. Then the
equations become
D1 = (g11-1)b1 X1 + g21b2X2 + g31b3X3 + U1
D2 = g12b1 X1 + (g22-1)b2X2 + g32b3X3 + U2
D3 = g13b1 X1 + g23b2X2 + (g33-1)b3X3 + U3,
Now for the statistical assumptions. To use regression
methods for this problem, it is needed to assume that
the U's have mean 0 and are uncorrelated with X1-X3.
This gives a quick solution to the problem; just run
regression WITHOUT CONSTANT TERMS of the D's on X1-X3.
The three regression will add up to 0. If regression
with constant terms is used, this removes the assumption
that the U's have mean 0, and they will still add up.
It turns out in this problem that the use of the other
X's will not change the estimates of the coefficients
without stronger assumptions. It is only the other
X's in this D-model which give overidentification.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
