1

Limitations of simple regression model:

Simultaneous equations

2

Causal model with reciprocal effects

D P

U1WI U2

+

-

P = priceD = demandI = IncomeW = Wages

3

Path Diagram

Y1

Y5Y4Y3

Y2

Y6

V2

* a

True value a=-.2

4

Example SEM, Monte Carlo data

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')

/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 1.5566 0.3683 0.4155 -0.0683 0.0235 0.0847 0.9376

n = 500

5

Standard OLS regression

When ignoring simultaneous equations, i.e. OLS:

MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS

V2 =V2 = .335*V1 + -.011*V4 + .312*V6 + 1.000 D2 .028 .030 .018 11.899 -.365 17.050

6

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 1.5566 0.3683 0.4155 -0.0683 0.0235 0.0847 0.9376 /END

SEM analysis

7

Simultaneous Equations

MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS

V1 =V1 = .863*V2 + .512*V3 + .512*V5 + 1.000 D1 .046 .019 .021 18.757 26.765 24.445

V2 =V2 = -.146*V1 + .471*V4 + .489*V6 + 1.000 D2 .058 .059 .028 -2.541 7.983 17.230

CHI-SQUARE = 0.213 BASED ON 3 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.97544