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.

$ontext
 
 
    Variance estimator for an MLE
 
    Erwin Kalvelagen July 2008
 
    Reference:
       William H. Greene, "Econometric Analysis", 5th edition, 2003
  
$offtext
  
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
$offecho
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.