86 Greene • Econometric Analysis, Seventh Edition
Applications
1. By adding the share equations vertically, we find the restrictions
1 + 2 + 3 = 1
11 + 12 + 13 = 0
By inserting these in the three share equations, we find
S1 = 1 + 11lnp1 + 12lnp2 − 11lnp3 − 12lnp3 + y1lnY + 1
For the cost function, making the substitutions for 3, 13, 23, 33, and y3 produces
lnC = + 1(lnp1 − lnp3) + 2(lnp2 − lnp3)
Chapter 10 Systems of Equations 87
The least squares estimates are shown below. Estimated standard errors appear in parentheses.
Variable Cost Function Capital Share Labor Share
One 51.32 (45.91) -.0174 (.4697) .2172 (.2408)
ln(pk/pf) -21.74 (20.14) .2380 (.1045) .0033 (.0534)
The estimates do not even come close to satisfying the cross equation restrictions. The parameters in
The results of estimation of the system by direct maximum likelihood are shown. The convergence
criterion is the value of Belsley (discussed near the end of Section 5.5). The value shown below is
gH−1g where g is the gradient and H is the Hessian of the log likelihood.
Iteration 0, F=46.76391, ln*S*= -7.514268, = 2.054399
Residual covariance matrix
Cost Capital Labor
Cost .0145572
Coefficient Estimate Std. Error
-6.41878 .6637
88 Greene • Econometric Analysis, Seventh Edition
k l f
0.01115 k
To test the hypothesis that yi = 0, we reestimate the model without the interaction terms between lnY
and the prices in the cost function and without lnY in the factor share equations. The iterations for this
restricted model are shown below.
Iter.= 0, F = 46.76391, log|S|= −7.514268, = 1.912223
Since we are interested only in the test statistic, we have not listed the parameter estimates. The test
?===========================================
? Application 10.2
?===========================================
? a. Separate regressions and aggregation test.
? This saves the residuals to be used later.
CALC ; SS1=0 $
MATRIX ; EOLS = Init(20,10,0) $
PROCEDURE $
+—————————————————-+
| Residuals Sum of squares = 143205.9 |
| Standard error of e = 91.78167 |
| Fit R-squared = .9213540 |
+—————————————————-+
Chapter 10 Systems of Equations 89
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .11928083 .02583417 4.617 .0002 4333.84500
| Residuals Sum of squares = 158093.3 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .17485602 .07419805 2.357 .0307 1971.82500
+—————————————————-+
| Residuals Sum of squares = 13216.59 |
| Standard error of e = 27.88272 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .02655119 .01556610 1.706 .1063 1941.32500
+—————————————————-+
| Residuals Sum of squares = 2997.444 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .07794782 .01997330 3.903 .0011 693.210000
+—————————————————-+
| Residuals Sum of squares = 1396.836 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
90 Greene • Econometric Analysis, Seventh Edition
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .13145484 .03117234 4.217 .0006 419.865000
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .08752720 .06562593 1.334 .1999 149.790000
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .05289413 .01570650 3.368 .0037 670.910000
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .07538794 .03395227 2.220 .0403 333.650000
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .00457343 .02716079 .168 .8683 70.9210000
Chapter 10 Systems of Equations 91
+—————————————————-+
| Ordinary least squares regression |
| LHS=I Mean = 145.9582 |
| Standard deviation = 216.8753 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .11556216 .00583571 19.803 .0000 1081.68110
Constant| -42.7143694 9.51167603 -4.491 .0000
? b. Aggregation test
REGRESS ; LHS = I ; RHS = F,C,one $
+————————————+
| Listed Calculator Results |
+————————————+
FSTAT = 5.131854
FC = 1.551534
? c. SUR model
NAMELIST ; X1=F1,C1,one $
NAMELIST ; X2=F2,C2,one $
NAMELIST ; X3=F3,C3,one $
SURE ; Lhs = Y ; Eq1=X1;Eq2=X2;Eq3=X3;Eq4=X4;Eq5=X6;Eq6=X6
Criterion function for GLS is log likelihood.
+—————————————————-+
| Estimates for equation: I1 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
92 Greene • Econometric Analysis, Seventh Edition
+—————————————————-+
| Estimates for equation: I2 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
+—————————————————-+
| Estimates for equation: I3 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F3 | .03425481 .00925706 3.700 .0002 1941.32500
+—————————————————-+
| Estimates for equation: I4 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
+—————————————————-+
| Estimates for equation: I5 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
+—————————————————-+
| Estimates for equation: I6 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+—————————————————-+
| Estimates for equation: I7 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
Chapter 10 Systems of Equations 93
+—————————————————-+
| Estimates for equation: I8 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+—————————————————-+
| Estimates for equation: I9 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
+—————————————————-+
| Estimates for equation: I10 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F10 | -.01695668 .01550963 -1.093 .2743 70.9210000
d. Aggregation test according to (10-15)
; list ; AggF = 1/27 * d'<vd>d $
Matrix AGGF has 1 rows and 1 columns.
1
+————–
1| 98.53777
+————————————+
| Listed Calculator Results |
+————————————+
Result = 1.551534
? e. Using separate OLS regressions, compute LM statistic.
? OLS residuals were saved in matrix EOLS earlier.
MATRIX ; VEOLS = 1/20*EOLS’EOLS
94 Greene • Econometric Analysis, Seventh Edition
+————————————+
| Listed Calculator Results |
+————————————+
? Constrained Sur model with one coefficient vector.
? This is the unconstrained model in (10-19)-(10-21)
SAMPLE ; 1 – 200 $
REGRESS; Lhs = I ; Rhs = F,C,one $
+—————————————————-+
| Ordinary least squares regression |
| LHS=I Mean = 145.9582 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
F | .11556216 .00583571 19.803 .0000 1081.68110
+————————————————–+
| Groupwise Regression Models |
| Estimator = 2 Step GLS |
+——–+————–+—————-+——–+——–+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|
+——–+————–+—————-+——–+——–+
F | .10806238 .00241169 44.808 .0000
+—————————————————-+
| Ordinary least squares regression |
| LHS=I Mean = 6.993136 |
Chapter 10 Systems of Equations 95
| Standard error of e = 7.703521 |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X|
?=========================================================
? Application 10.3 Simultaneous Equations
?=========================================================
? Read the data
? For convenience, rename the variables so they correspond
? to the example in the text.
sample ; 1 – 204 $
create ; ct=realcons$
? Estimate equations by 2sls and save coefficients with
? the names used in the example.
2sls ; lhs = ct ; rhs=one,yt,ct1 ; inst = xt $
+—————————————————-+
| Two stage least squares regression |
| LHS=CT Mean = 3008.995 |
| Standard deviation = 1456.900 |
+—————————————————-+
| Instrumental Variables:
|ONE GT RT CT1 YT1
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
96 Greene • Econometric Analysis, Seventh Edition
+—————————————————-+
| Two stage least squares regression |
| LHS=IT Mean = 654.5296 |
+—————————————————-+
| Instrumental Variables:
|ONE GT RT CT1 YT1
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
Constant| -300.699429 125.980850 -2.387 .0170
?
? Create the coefficients of the reduced form. We only need the parts
? for the dynamics. These are in the second half of the example.
calc ; a=1-a1-b2 $
?
? Construct the matrix that governs the dynamics of the system. Note that
? the I equation is static. It is a function of y(t-1) and c(t-1) but not
+————————————+
| Listed Calculator Results |
+————————————+
C11 = .996253
C12 = .061967
Matrix ROOTS has 2 rows and 2 columns.
1 2
+————————————+
| Listed Calculator Results |
+————————————+
DOMROOT = 1.029596
Chapter 10 Systems of Equations 97
+—————————————————-+
| Estimates for equation: CT |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
| Estimates for equation: IT |
| InstVar/GLS least squares regression |
+—————————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+