Download LINEAR STRUCTURAL RELATIONS

Limitations of simple regression
model:
Simultaneous equations
1
Causal model with reciprocal
effects
U1
P = price
D = demand
I = Income
W = Wages
W
I
D
+
-
U2
P
2
True value
a=-.2
Path Diagram
V2
*a
Y1
Y3
Y2
Y4
Y5
Y6
3
Example SEM, Monte Carlo data
/MATRIX
2.5123
0.9345 0.5414
1.4768 0.4544
2.1110 0.6925
1.5067 0.5093
0.3683 0.4155
1.5910
1.4842 2.1037
0.5278 1.4727 1.5566
-0.0683 0.0235 0.0847 0.9376
n = 500
Example SEM, Monte Carlo DAta
Data generating process (Matlab code):
n = 500;
B = [0 .9 .5 0 .5 0;
-.2 0 0 .5 0 .5;
0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0 0 0]
IB = inv( eye(6,6)-B);
PHI = [.2 0 0 0 0 0;
0 .2 0 0 0 0;
0 0 1 .4 0 0;
0 0 .4 1 .4 0;
0 0 0 .4 1 0;
0 0 0
0 0 1]
z = IB*sqrt(PHI)*randn(6, n);
SIG = cov(z')
4
Standard OLS regression
When ignoring simultaneous equations, i.e. OLS:
MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS
V2
=V2 =
.335*V1
.028
11.899
+ -.011*V4
.030
-.365
+ .312*V6
.018
17.050
+ 1.000 D2
5
SEM analysis
EQS analysis:
GOODNESS OF FIT SUMMARY
/TITLE
example of SEM
/SPECIFICATIONS
CASES = 500 ; VAR = 6;
/EQUATIONS
V1 =
.5*V2 + *V3 + *V5 +D1 ;
V2 =
-.5*V1 + *V4 + *V6 + D2 ;
/VARIANCES
D1 = *;
D2 = *;
V3 = *; V4 = *; V5 = *; V6=*;
/COVARIANCES
V3 to V6 = *;
/MATRIX
2.5123
0.9345
0.5414
1.4768
0.4544
1.5910
2.1110
0.6925
1.4842
2.1037
1.5067
0.5093
0.5278
1.4727
0.3683
0.4155
-0.0683
0.0235
0.9376
/END
1.5566
0.0847
6
Simultaneous Equations
MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS
V1
=V1
=
.863*V2
.046
18.757
+
.512*V3
.019
26.765
V2
=V2
=
-.146*V1
.058
-2.541
+
.471*V4
.059
7.983
+
.512*V5
.021
24.445
+ 1.000 D1
+
+ 1.000 D2
.489*V6
.028
17.230
CHI-SQUARE =
0.213 BASED ON
3 DEGREES OF FREEDOM
PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS
0.97544
7