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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?
Dick
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
>
> ----------------------------------------------------------------
> Kenneth Benoit
> [EMAIL PROTECTED]
> Nov 26, 1999
>
> 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 ³ Yi ³ 1.0 "i
> 1 = X0 + X4 + X5, 0 ³ Xi ³ 1.0 "i
> 1 = g11 + g12 + g13, 0 ³ gij ³ 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.
>
> -----------------------------------------------------------------------
> File translated fromTEXby TTH,version 2.56.
> On 26 Nov 1999, 14:00.
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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?
<p>Dick
<p>Kenneth Benoit wrote:
<blockquote TYPE=CITE>
<pre>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
<a href="http://benoit.tcd.ie">http://benoit.tcd.ie
</a>Department of Political Science <a
href="mailto:[EMAIL PROTECTED]">mailto:[EMAIL PROTECTED]
</a>Trinity
College
Tel: 353-1-608-2491
Dublin 2,
Ireland
Fax: 353-1-677-0546</pre>
<hr WIDTH="90%" SIZE=4>Kenneth Benoit
<br><tt>[EMAIL PROTECTED]</tt>
<br>Nov 26, 1999
<p><b>Consider a system where:</b>
<center><table BORDER=0 >
<tr>
<td>
<table ALIGN=LEFT >
<tr>
<td ALIGN=CENTER NOWRAP></td>
<td ALIGN=CENTER NOWRAP>
<table>
<tr>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>Y<sub>1</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>= </td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>X<sub>0</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>(1<font face="symbol">-g</font><sub>11</sub>)<font
face="symbol">b</font><sub>1</sub>
X<sub>1</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>21</sub><font
face="symbol">b</font><sub>2</sub>X<sub>2</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>31</sub><font
face="symbol">b</font><sub>3</sub>X<sub>3</sub></td>
</tr>
</table>
</td>
</tr>
<tr>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>Y<sub>2</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>= </td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>X<sub>4</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>12</sub><font
face="symbol">b</font><sub>1</sub>
X<sub>1</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>(1<font face="symbol">-g</font><sub>22</sub>)<font
face="symbol">b</font><sub>2</sub>X<sub>2</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>32</sub><font
face="symbol">b</font><sub>3</sub>X<sub>3</sub></td>
</tr>
</table>
</td>
</tr>
<tr>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>Y<sub>3</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>= </td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>X<sub>5</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>13</sub><font
face="symbol">b</font><sub>1</sub>
X<sub>1</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP><font face="symbol">g</font><sub>23</sub><font
face="symbol">b</font><sub>2</sub>X<sub>2</sub></td>
</tr>
</table>
</td>
<td ALIGN=CENTER>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>+ </td>
</tr>
</table>
</td>
<td ALIGN=RIGHT>
<table BORDER=0 >
<tr>
<td ALIGN=CENTER NOWRAP>(1<font face="symbol">-g</font><sub>33</sub>)<font
face="symbol">b</font><sub>3</sub>X<sub>3</sub></td>
</tr>
</table>
</td>
</tr>
</table>
</td>
<td ALIGN=CENTER NOWRAP></td>
</tr>
</table>
</td>
</tr>
</table></center>
<p>and:
<p>1 = Y<sub>1</sub> + Y<sub>2</sub> + Y<sub>3</sub>, 0 <font
face="symbol">³</font>
Y<sub>i</sub> <font face="symbol">³</font> 1.0 <font face="symbol">"</font>i
<br>1 = X<sub>0</sub> + X<sub>4</sub> + X<sub>5</sub>, 0 <font
face="symbol">³</font>
X<sub>i</sub> <font face="symbol">³</font> 1.0 <font face="symbol">"</font>i
<br>1 = <font face="symbol">g</font><sub>11</sub> + <font
face="symbol">g</font><sub>12</sub>
+ <font face="symbol">g</font><sub>13</sub>, 0 <font face="symbol">³</font>
<font face="symbol">g</font><sub>ij</sub> <font face="symbol">³</font>
1.0 <font face="symbol">"</font>i,j
<br>1 = <font face="symbol">g</font><sub>21</sub> + <font
face="symbol">g</font><sub>22</sub>
+ <font face="symbol">g</font><sub>33</sub>
<br>1 = <font face="symbol">g</font><sub>31</sub> + <font
face="symbol">g</font><sub>32</sub>
+ <font face="symbol">g</font><sub>33</sub>
<p><b>GOAL:</b> To estimate <font face="symbol">g</font>'s and <font
face="symbol">b</font>'s.
Problems: overidentification; effects of the data items and some of the
parameters summing to 1 which I still don't fully understand.
<p><b>Background:</b> This is for a voting transition study in Italy, where
the <font face="symbol">b</font>'s represent a the proportion of voters
following a rational proximity model, and the <font face="symbol">g</font>'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.
<p>Possible ways to simplify:
<ul>
<li>
Set <font face="symbol">g</font><sub>ij</sub>'s to constants before estimation.</li>
<li>
Set <font face="symbol">g</font><sub>ij</sub> = <font
face="symbol">g</font><sup>*</sup>
<font face="symbol">"</font> i,j.</li>
<li>
Set <font face="symbol">b</font><sub>3</sub> = 1.</li>
<li>
Set <font face="symbol">b</font><sub>1</sub> = <font
face="symbol">b</font><sub>2</sub>.</li>
</ul>
<hr><font size=-1>File translated fromT<sub>E</sub>Xby <a
href="http://hutchinson.belmont.ma.us/tth/">T<sub>T</sub>H</a>,version
2.56.</font>
<br><font size=-1>On 26 Nov 1999, 14:00.</font></blockquote>
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