[Gretl-users] Re: Questions in Simultaneous Equations Systems

[Allin Cottrell]

On Wed, 3 May 2006, Marta Regulez wrote:

> I am teaching now a new course in Simultaneous Equations Models, 
> and the new features that are in Gretl is being very usefu for 
> my teaching and for the students.

That's good to hear.  I'm copying the gretl-users mailing list on 
my reply to your questions below, since these questions may come 
up for others too.

> I have used the klein.inp script to estimate "Klein Model I" and 
> I have the following questions (I couldn´t find some hint in the 
> guide):
>
> 1) Once you estimate by LIML, the program shows the Smallest 
> eigenvalue "lambda" and the overidentification LR test that I 
> thought was the Anderson-Rubin test
>
> T(lambda - 1)
>
> For example for the Consumption Equation it is obtained
>
> Smallest eigenvalue = 1,49875
>  Contraste de sobreidentificación LR:
>    Chi-cuadrado(4) = 8,4972 con valor p  0,0749722
>
> But if you calculate T(lambda - 1)= 21( 1,49875 - 1) = 10, 47375 
> that is not the value given for the LR test.

I am going by the presentation of the Anderson-Rubin test as given 
in Davidson and MacKinnon, "Econometric Theory and Methods" (ETM)
chapter 12.  They write the LR test statistic as

T log(lambda)

[in their notation, n log \hat{\kappa}], and state that it is 
asympotically distributed as chi-square with degrees of freedom 
equal to the number of overidentifying restrictions.  T(lambda - 
1) is a reasonable approximation to T log(lambda) for lambda close 
to 1.

> 2) In the output obtained with ols, tsls, 3sls, fiml, and liml, 
> What are the following items?
>
> Matriz de covarianzas cruzada residual
> (correlaciones por encima de la diagonal principal)
>
>       2,1041      (0,748)      (0,247)
>       3,8790       12,771      (0,804)
>      0,48169       3,8575       1,8011
>
> logaritmo del determinante = 0,366633

This is the cross-equation variance-covariance matrix, with 
correlation coefficients in parentheses.  For example, the (2,1) 
entry, 3.8790, gives the covariance of the residuals from 
equations 1 and 2.  The (1,2) entry, 0.748, gives the correlation 
coefficient for the residuals from equations 1 and 2.

> 3) Finally, the overidentification test given with 3sls for the 
> whole system, refered as Hansen-Sargan overidentification test, 
> how is it calculated?

This is the minimized value of the 3sls criterion function (that 
is, the analog to the sum of squared residuals in OLS estimation).
I think it's sometimes called Hansen's J statistic.  For the 
specific calculation, please see pages 525 and 532 of ETM.

Systems estimation in gretl needs to be better documented.  But 
the gretl code is closely based on the presentation in chapter 12 
of Davidson and MacKinnon's ETM, so that is the best place to look 
for clarification at present.

Allin Cottrell