Example 14.8: Nonlinear FIML Estimation

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 10^c₂z₄(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 c_i'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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