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  Nonlinear least squares.

  Example: Estimation of a CES production function

  Data set: Table 22.4, page 724 of Griffiths, Hill and Judge,
     LEARNING AND PRACTICING ECONOMETRICS, Wiley, 1993.

  Erwin Kalvelagen, 2000

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set i 'observations' /i1*i30/;
set j 'parameters'   /L,K,Q/;

table data(i,j)
          L          K           Q
i1      0.228      0.802      0.256918
i2      0.258      0.249      0.183599
i3      0.821      0.771      1.212883
i4      0.767      0.511      0.522568
i5      0.495      0.758      0.847894
i6      0.487      0.425      0.763379
i7      0.678      0.452      0.623130
i8      0.748      0.817      1.031485
i9      0.727      0.845      0.569498
i10     0.695      0.958      0.882497
i11     0.458      0.084      0.108827
i12     0.981      0.021      0.026437
i13     0.002      0.295      0.003750
i14     0.429      0.277      0.461626
i15     0.231      0.546      0.268474
i16     0.664      0.129      0.186747
i17     0.631      0.017      0.020671
i18     0.059      0.906      0.100159
i19     0.811      0.223      0.252334
i20     0.758      0.145      0.103312
i21     0.050      0.161      0.078945
i22     0.823      0.006      0.005799
i23     0.483      0.836      0.723250
i24     0.682      0.521      0.776468
i25     0.116      0.930      0.216536
i26     0.440      0.495      0.541182
i27     0.456      0.185      0.316320
i28     0.342      0.092      0.123811
i29     0.358      0.485      0.386354
i30     0.162      0.934      0.279431
;


parameters
    L(i)         'labor'
    K(i)         'capital'
    Q(i)         'output'
;

L(i) = data(i,'L');
K(i) = data(i,'K');
Q(i) = data(i,'Q');


variables
    gamma        'log of efficiency parameter'
    delta        'distribution parameter'
    rho          'substitution parameter'
    eta          'homogeneity parameter'
    residual(i)  'error term'
    sse          'sum of squared errors'
;

equations
    fit(i)     'the nonlinear model'
    obj        'objective'
;


obj..     sse =e= sum(i, sqr(residual(i)));
fit(i)..  log(Q(i)) =e=
             gamma - (eta/rho)*log[delta*L(i)**(-rho) + (1-delta)*K(i)**(-rho)]
             + residual(i);

* initial values
rho.l=1;
delta.l=0.5;
gamma.l=1;
eta.l=1;

model nls /obj,fit/;

solve nls minimizing sse using nlp;

display gamma.l, delta.l, rho.l, eta.l, sse.l;