Example 19.8 Nonlinear FIML Estimation :: SAS/ETS(R) 13.1 User's Guide

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