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 The MODEL Procedure

## Example 14.8: Nonlinear FIML Estimation

The data and model for this example were obtained from Bard (1974, p.133-138). The example is a two-equation econometric model used by Bodkin and Klein to fit U.S production data for the years 1909-1949. The model is the following:

g1 = c1 10c2z4(c5 z-c41 + (1- c5) z-c42)-c3/c4 - z3 = 0
g2 = [c5/(1-c5)](z1/z2)(-1 - c4) - z5 = 0

where z1 is capital input, z2 is labor input, z3 is real output, z4 is time in years with 1929 as year zero, and z5 is the ratio of price of capital services to wage scale. The ci's are the unknown parameters. z1 and z2 are considered endogenous variables. A FIML estimation is performed.

```   data bodkin;
input z1 z2 z3 z4 z5;
datalines;
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
;

proc model data=bodkin;
parms c1-c5;
endogenous z1 z2;
exogenous z3 z4 z5;

eq.g1 = c1 * 10 **(c2 * z4) * (c5*z1**(-c4)+
(1-c5)*z2**(-c4))**(-c3/c4) - z3;
eq.g2 = (c5/(1-c5))*(z1/z2)**(-1-c4) -z5;

fit g1 g2 / fiml ;
run;
```

When FIML estimation is selected, the log likelihood of the system is output as the objective value. The results of the estimation are show in Output 14.8.1.

Output 14.8.1: FIML Estimation Results for U.S. Production Data

 The MODEL Procedure

 Nonlinear FIML Summary of Residual Errors Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq g1 4 37 0.0529 0.00143 0.0378 g2 1 40 0.0173 0.000431 0.0208

 Nonlinear FIML Parameter Estimates Parameter Estimate Approx Std Err t Value Approx Pr > |t| c1 0.58395 0.0218 26.76 <.0001 c2 0.005877 0.000673 8.74 <.0001 c3 1.3636 0.1148 11.87 <.0001 c4 0.473688 0.2699 1.75 0.0873 c5 0.446748 0.0596 7.49 <.0001

 Number of Observations Statistics for System Used 41 Log Likelihood 110.7773 Missing 0

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