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 The MODEL Procedure

Example 14.2: A Consumer Demand Model

This example shows the estimation of a system of nonlinear consumer demand equations based on the translog functional form using seemingly unrelated regression (SUR). Expenditure shares and corresponding normalized prices are given for three goods.

Since the shares add up to one, the system is singular; therefore, one equation is omitted from the estimation process. The choice of which equation to omit is arbitrary. The parameter estimates of the omitted equation (share3) can be recovered from the other estimated parameters. The nonlinear system is first estimated in unrestricted form.

```   title1 'Consumer Demand--Translog Functional Form';
title2 'Nonsymmetric Model';
proc model data=tlog1;
var share1 share2 p1 p2 p3;
parms a1 a2 b11 b12 b13 b21 b22 b23 b31 b32 b33;
bm1 = b11 + b21 + b31;
bm2 = b12 + b22 + b32;
bm3 = b13 + b23 + b33;
lp1 = log(p1);
lp2 = log(p2);
lp3 = log(p3);
share1 = ( a1 + b11 * lp1 + b12 * lp2 + b13 * lp3 ) /
( -1 + bm1 * lp1 + bm2 * lp2 + bm3 * lp3 );
share2 = ( a2 + b21 * lp1 + b22 * lp2 + b23 * lp3 ) /
( -1 + bm1 * lp1 + bm2 * lp2 + bm3 * lp3 );
fit share1 share2
start=( a1 -.14 a2 -.45 b11 .03 b12 .47 b22 .98 b31 .20
b32 1.11 b33 .71 ) / outsused = smatrix sur;
run;
```
A portion of the printed output produced in the preceding example is shown in Output 14.2.1 .

Output 14.2.1: Estimation Results from the Unrestricted Model

 Consumer Demand--Translog Functional Form Nonsymmetric Model

 The MODEL Procedure

 Model Summary Model Variables 5 Parameters 11 Equations 2 Number of Statements 8

 Model Variables share1 share2 p1 p2 p3 Parameters a1(-0.14) a2(-0.45) b11(0.03) b12(0.47) b13 b21 b22(0.98) b23 b31(0.2) b32(1.11) b33(0.71) Equations share1 share2

 Consumer Demand--Translog Functional Form Nonsymmetric Model

 The MODEL Procedure

 The 2 Equations to Estimate share1 = F(a1, b11, b12, b13, b21, b22, b23, b31, b32, b33) share2 = F(a2, b11, b12, b13, b21, b22, b23, b31, b32, b33)

 NOTE: At SUR Iteration 2 CONVERGE=0.001 Criteria Met.

 Consumer Demand--Translog Functional Form Nonsymmetric Model

 The MODEL Procedure

 Nonlinear SUR Summary of Residual Errors Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq share1 5.5 38.5 0.00166 0.000043 0.00656 0.8067 0.7841 share2 5.5 38.5 0.00135 0.000035 0.00592 0.9445 0.9380

 Nonlinear SUR Parameter Estimates Parameter Estimate Approx Std Err t Value Approx Pr > |t| a1 -0.14881 0.00225 -66.08 <.0001 a2 -0.45776 0.00297 -154.29 <.0001 b11 0.048382 0.0498 0.97 0.3379 b12 0.43655 0.0502 8.70 <.0001 b13 0.248588 0.0516 4.82 <.0001 b21 0.586326 0.2089 2.81 0.0079 b22 0.759776 0.2565 2.96 0.0052 b23 1.303821 0.2328 5.60 <.0001 b31 0.297808 0.1504 1.98 0.0550 b32 0.961551 0.1633 5.89 <.0001 b33 0.8291 0.1556 5.33 <.0001

 Number of Observations Statistics for System Used 44 Objective 1.7493 Missing 0 Objective*N 76.9697

The model is then estimated under the restriction of symmetry (bij=bji).

Hypothesis testing requires that the S matrix from the unrestricted model be imposed on the restricted model, as explained in "Tests on Parameters" in this chapter. The S matrix saved in the data set SMATRIX is requested by the SDATA= option.

A portion of the printed output produced in the following example is shown in Output 14.2.2.

```   title2 'Symmetric Model';
proc model data=tlog1;
var share1 share2 p1 p2 p3;
parms a1 a2 b11 b12 b22 b31 b32 b33;
bm1 = b11 + b12 + b31;
bm2 = b12 + b22 + b32;
bm3 = b31 + b32 + b33;
lp1 = log(p1);
lp2 = log(p2);
lp3 = log(p3);
share1 = ( a1 + b11 * lp1 + b12 * lp2 + b31 * lp3 ) /
( -1 + bm1 * lp1 + bm2 * lp2 + bm3 * lp3 );
share2 = ( a2 + b12 * lp1 + b22 * lp2 + b32 * lp3 ) /
( -1 + bm1 * lp1 + bm2 * lp2 + bm3 * lp3 );
fit share1 share2
start=( a1 -.14 a2 -.45 b11 .03 b12 .47 b22 .98 b31 .20
b32 1.11 b33 .71 ) / sdata=smatrix sur;
run;
```

A chi-square test is used to see if the hypothesis of symmetry is accepted or rejected. (Oc-Ou) has a chi-square distribution asymptotically, where Oc is the constrained OBJECTIVE*N and Ou is the unconstrained OBJECTIVE*N. The degrees of freedom is equal to the difference in the number of free parameters in the two models.

In this example, Ou is 76.9697 and Oc is 78.4097, resulting in a difference of 1.44 with 3 degrees of freedom. You can obtain the probability value by using the following statements:

```   data _null_;
/* reduced-full, nrestrictions */
p = 1-probchi( 1.44, 3 );
put p=;
run;
```

The output from this DATA step run is `P=0.6961858724'. With this probability you cannot reject the hypothesis of symmetry. This test is asymptotically valid.

Output 14.2.2: Estimation Results from the Restricted Model

 Consumer Demand--Translog Functional Form Symmetric Model

 The MODEL Procedure

 The 2 Equations to Estimate share1 = F(a1, b11, b12, b22, b31, b32, b33) share2 = F(a2, b11, b12, b22, b31, b32, b33)

 Consumer Demand--Translog Functional Form Symmetric Model

 The MODEL Procedure

 Nonlinear SUR Summary of Residual Errors Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq share1 4 40 0.00166 0.000041 0.00644 0.8066 0.7920 share2 4 40 0.00139 0.000035 0.00590 0.9428 0.9385

 Nonlinear SUR Parameter Estimates Parameter Estimate Approx Std Err t Value Approx Pr > |t| a1 -0.14684 0.00135 -108.99 <.0001 a2 -0.4597 0.00167 -275.34 <.0001 b11 0.02886 0.00741 3.89 0.0004 b12 0.467827 0.0115 40.57 <.0001 b22 0.970079 0.0177 54.87 <.0001 b31 0.208143 0.00614 33.88 <.0001 b32 1.102415 0.0127 86.51 <.0001 b33 0.694245 0.0168 41.38 <.0001

 Number of Observations Statistics for System Used 44 Objective 1.7820 Missing 0 Objective*N 78.4097

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