Thursday, July 31, 2008

Hessian via Convert

GAMS has new facilities to retrieve the Hessian matrix (matrix of second derivatives). Here we use Convert to produce a gdx file with the Hessian which we use to estimate the Variance of a Maximum Likelihood Estimation result.

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Variance estimator for an MLE

Erwin Kalvelagen July 2008

Reference:
William H. Greene, "Econometric Analysis", 5th edition, 2003

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set i 'cases' /i1*i20/;

table data(i,*)

Y E
i1 20.5 12
i2 31.5 16
i3 47.7 18
i4 26.2 16
i5 44.0 12
i6 8.28 12
i7 30.8 16
i8 17.2 12
i9 19.9 10
i10 9.96 12
i11 55.8 16
i12 25.2 20
i13 29.0 12
i14 85.5 16
i15 15.1 10
i16 28.5 18
i17 21.4 16
i18 17.7 20
i19 6.42 12
i20 84.9 16
;

parameter y(i),x(i);
y(i) = data(i,'y');
x(i) = data(i,'E');

scalar Var2Greene 'Variance estimate number two from Greene' / 46.16337 /;

variable b,L;
equation loglik;

loglik.. L =e= sum(i,log(b+x(i))) + sum(i,y(i)/(b+x(i)));


*
* estimation
*
model estimation /loglik/;
solve estimation minimizing L using nlp;
display "estimate",b.l;


*
* get hessian
*
option nlp=convert;
$onecho > convert.opt;
hessian
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estimation.optfile=1;
solve estimation minimizing L using nlp;

*
* gams cannot add elements at runtime so we declare the necessary elements here
*
set dummy /e1,x1/;

parameter h(*,*,*) '-hessian';
execute_load "hessian.gdx",h;
display h;

*
* inverting a 1x1 matrix is not very difficult
*
scalar v 'variance';
v = -1/h('e1','x1','x1');
display v,var2greene;

abort$(abs(v-var2greene)>1.0e-5) "Accuracy check failed";


Note: if you get a message
*** Msg=Error loading library c:\progra~1\gams\g2d: The specified module could not be found.
then copy gd.dll from the GAMS system directory to c:\progra~1\gams. This bug is already fixed.