/* An Example of Nonlinear Least Squares Using the CES Production
   Function.  See the manual SAS/STAT USER'S GUIDE, VOL. 2,
   GLM-VARCOMP (VERSION 6, FOURTH EDITION), p. 1159  */

title 'CES MODEL: LOGQ = B0 + A*LOG(D*L**R + (1-D)*K**R)';

data ces;
  input l k logq @@;
  cards;
.228 .802 -1.359 .258 .249 -1.695
.821 .771   .193 .767 .511  -.649
.495 .758  -.165 .487 .425  -.270
.678 .452  -.473 .748 .817   .031
.727 .845  -.563 .695 .958  -.125
.458 .084 -2.218 .981 .021 -3.633
.002 .295 -5.586 .429 .277  -.773
.231 .546 -1.315 .664 .129 -1.678
.631 .017 -3.879 .059 .906 -2.301
.811 .223 -1.377 .758 .145 -2.270
.050 .161 -2.539 .823 .006 -5.150
.483 .836  -.324 .682 .521  -.253
.116 .930 -1.530 .440 .495  -.614
.456 .185 -1.151 .342 .092 -2.089
.358 .485  -.951 .162 .934 -1.275
;

/* Use the Gauss-Newton method to estimate the CES production function */

proc nlin data=ces;
   parms b0=1 a=-1 d=.5 r=-1;
   lr=l**r;
   kr=k**r;
   z=d*lr+(1-d)*kr;
   model logq=b0+a*log(z);
   der.b0=1;
   der.a=log(z);
   der.d=(a/z)*(lr-kr);
   der.r=(a/z)*(d*log(l)*lr+(1-d)*log(k)*kr);
run;

/* Let's use multiple starting points to assure ourselves that we have
   reached global minimum  */

proc nlin data=ces best=8;
   parms b0=0 to 2 by 0.5 a=-2 to 0 by 0.5 d=0 to 1 by 0.5 r=-2 to 0 by 0.5;
   lr=l**r;
   kr=k**r;
   z=d*lr+(1-d)*kr;
   model logq=b0+a*log(z);
   der.b0=1;
   der.a=log(z);
   der.d=(a/z)*(lr-kr);
   der.r=(a/z)*(d*log(l)*lr+(1-d)*log(k)*kr);
run;

/* Let's use the secant method (DUD) by using proc nlin but without
   specifying the first derivatives.  We should get roughly the same
   results obtained in the Gauss-Newton method.  */

proc nlin data=ces;
   parms b0=1 a=-1 d=.5 r=-1;
   lr=l**r;
   kr=k**r;
   z=d*lr+(1-d)*kr;
   model logq=b0+a*log(z);
run;

proc nlin data=ces best=8;
   parms b0=0 to 2 by 0.5 a=-2 to 0 by 0.5 d=0 to 1 by 0.5 r=-2 to 0 by 0.5;
   lr=l**r;
   kr=k**r;
   z=d*lr+(1-d)*kr;
   model logq=b0+a*log(z);
run;