The MODEL Procedure

Example 19.8 Nonlinear FIML Estimation

The data and model for this example were obtained from Bard (1974, pp. 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:

\[ g_{1} = c_{1} 10^{c_{2}z_{4}}(c_{5} z^{-c_{4}}_{1} + (1- c_{5}) z^{-c_{4}}_{2})^{-c_{3}/c_{4}} - z_{3} = 0 \]
\[ g_{2} = [c_{5}/(1-c_{5})](z_{1}/z_{2})^{(-1 - c_{4})} - z_{5} = 0 \]

where ${z_{1}}$ is capital input, ${z_{2}}$ is labor input, ${z_{3}}$ is real output, ${z_{4}}$ is time in years with 1929 as year zero, and ${z_{5}}$ is the ratio of price of capital services to wage scale. The ${c_{i}}$’s are the unknown parameters. ${z_{1}}$ and ${z_{2}}$ are considered endogenous variables. A FIML estimation is performed by using the following statements: 

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

   ... more lines ...   


title1 "Nonlinear FIML Estimation";

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 shown in Output 19.8.1.

Output 19.8.1: FIML Estimation Results for U.S. Production Data

Nonlinear FIML Estimation

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