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| Nonlinear Optimization Examples |
This example and notation are taken from Bard (1974). A two-equation model is used to fit U.S. production data for the years 1909-1949, where z1 is capital input, z2 is labor input, z3 is real output, z4 is time in years (with 1929 as the origin), and z5 is the ratio of price of capital services to wage scale.
proc iml;
z={ 1.33135 0.64629 0.4026 -20 0.24447,
1.39235 0.66302 0.4084 -19 0.23454,
1.41640 0.65272 0.4223 -18 0.23206,
1.48773 0.67318 0.4389 -17 0.22291,
1.51015 0.67720 0.4605 -16 0.22487,
1.43385 0.65175 0.4445 -15 0.21879,
1.48188 0.65570 0.4387 -14 0.23203,
1.67115 0.71417 0.4999 -13 0.23828,
1.71327 0.77524 0.5264 -12 0.26571,
1.76412 0.79465 0.5793 -11 0.23410,
1.76869 0.71607 0.5492 -10 0.22181,
1.80776 0.70068 0.5052 -9 0.18157,
1.54947 0.60764 0.4679 -8 0.22931,
1.66933 0.67041 0.5283 -7 0.20595,
1.93377 0.74091 0.5994 -6 0.19472,
1.95460 0.71336 0.5964 -5 0.17981,
2.11198 0.75159 0.6554 -4 0.18010,
2.26266 0.78838 0.6851 -3 0.16933,
2.33228 0.79600 0.6933 -2 0.16279,
2.43980 0.80788 0.7061 -1 0.16906,
2.58714 0.84547 0.7567 0 0.16239,
2.54865 0.77232 0.6796 1 0.16103,
2.26042 0.67880 0.6136 2 0.14456,
1.91974 0.58529 0.5145 3 0.20079,
1.80000 0.58065 0.5046 4 0.18307,
1.86020 0.62007 0.5711 5 0.18352,
1.88201 0.65575 0.6184 6 0.18847,
1.97018 0.72433 0.7113 7 0.20415,
2.08232 0.76838 0.7461 8 0.18847,
1.94062 0.69806 0.6981 9 0.17800,
1.98646 0.74679 0.7722 10 0.19979,
2.07987 0.79083 0.8557 11 0.21115,
2.28232 0.88462 0.9925 12 0.23453,
2.52779 0.95750 1.0877 13 0.20937,
2.62747 1.00285 1.1834 14 0.19843,
2.61235 0.99329 1.2565 15 0.18898,
2.52320 0.94857 1.2293 16 0.17203,
2.44632 0.97853 1.1889 17 0.18140,
2.56478 1.02591 1.2249 18 0.19431,
2.64588 1.03760 1.2669 19 0.19492,
2.69105 0.99669 1.2708 20 0.17912 };
The two-equation model in five parameters c1, ... ,c5 is
![g_1 & = & c_110^{c_2z_4}[c_5z_1^{-c_4} +
(1-c_5)z_2^{-c_4}]^{-c_3/c_4} - z_3 = 0 \g_2 & = & [\frac{c_5}{1-c_5}] ( \frac{z_1}{z_2}
)^{-1-c_4} -z_5 = 0](images/i11eq178.gif)
where the variables z1 and z2 are considered dependent (endogenous) and the variables z3, z4, and z5 are considered independent (exogenous).
Differentiation of the two equations g1 and g2 with
respect to the endogenous variables z1 and z2 yields
the Jacobian matrix 
for i=1,2
and j=1,2, where i corresponds to rows (equations) and
j corresponds to endogenous variables (refer to Bard 1974).
You must consider parameter sets for which the elements of the
Jacobian and the logarithm of the determinant cannot be computed.
In such cases, the function module must return a missing value.
start fiml(pr) global(z);
c1 = pr[1]; c2 = pr[2]; c3 = pr[3]; c4 = pr[4]; c5 = pr[5];
/* 1. Compute Jacobian */
lndet = 0 ;
do t= 1 to 41;
j11 = (-c3/c4) * c1 * 10 ##(c2 * z[t,4]) * (-c4) * c5 *
z[t,1]##(-c4-1) * (c5 * z[t,1]##(-c4) + (1-c5) *
z[t,2]##(-c4))##(-c3/c4 -1);
j12 = (-c3/c4) * (-c4) * c1 * 10 ##(c2 * z[t,4]) * (1-c5) *
z[t,2]##(-c4-1) * (c5 * z[t,1]##(-c4) + (1-c5) *
z[t,2]##(-c4))##(-c3/c4 -1);
j21 = (-1-c4)*(c5/(1-c5))*z[t,1]##( -2-c4)/ (z[t,2]##(-1-c4));
j22 = (1+c4)*(c5/(1-c5))*z[t,1]##( -1-c4)/ (z[t,2]##(-c4));
j = (j11 || j12 ) // (j21 || j22) ;
if any(j = .) then detj = 0.;
else detj = det(j);
if abs(detj) < 1.e-30 then do;
print t detj j;
return(.);
end;
lndet = lndet + log(abs(detj));
end;
Assuming that the residuals of the two equations are normally distributed, the likelihood is then computed as in Bard (1974). The following code computes the logarithm of the likelihood function:
/* 2. Compute Sigma */
sb = j(2,2,0.);
do t= 1 to 41;
eq_g1 = c1 * 10##(c2 * z[t,4]) * (c5*z[t,1]##(-c4)
+ (1-c5)*z[t,2]##(-c4))##(-c3/c4) - z[t,3];
eq_g2 = (c5/(1-c5)) * (z[t,1] / z[t,2])##(-1-c4) - z[t,5];
resid = eq_g1 // eq_g2;
sb = sb + resid * resid`;
end;
sb = sb / 41;
/* 3. Compute log L */
const = 41. * (log(2 * 3.1415) + 1.);
lnds = 0.5 * 41 * log(det(sb));
logl = const - lndet + lnds;
return(logl);
finish fiml;
There are potential problems in computing the power and log functions for an unrestricted parameter set. As a result, optimization algorithms that use line search will fail more often than algorithms that restrict the search area. For that reason, the NLPDD subroutine is used in the following code to maximize the log-likelihood function:
pr = j(1,5,0.001);
optn = {0 2};
tc = {. . . 0};
call nlpdd(rc, xr,"fiml", pr, optn,,tc);
print "Start" pr, "RC=" rc, "Opt Par" xr;
Part of the iteration history is shown in Output 11.7.1.
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