Example 17.2: Simultaneous Equations with Intercept

The CALIS Procedure

Example 17.2: Simultaneous Equations with Intercept

The demand-and-supply food example of Kmenta (1971, pp. 565, 582) is used to illustrate the use of PROC CALIS for the estimation of intercepts and coefficients of simultaneous equations. The model is specified by two simultaneous equations containing two endogenous variables Q and P and three exogenous variables D, F, and Y,

$Q_t(demand) = \alpha_1 + \beta_1 P_t + \gamma_1 D_t$

$Q_t(supply) = \alpha_2 + \beta_2 P_t + \gamma_2 F_t + \gamma_3 Y_t$

for t = 1, ... , 20.

The LINEQS statement requires that each endogenous variable appear on the left-hand side of exactly one equation. Instead of analyzing the system

$B^* eta = {{\Gamma}}{\xi}+ {\zeta}$

PROC CALIS analyzes the equivalent system

$eta = Beta + {{\Gamma}}{\xi}+ {\zeta}$

with B^* = I- B. This requires that one of the preceding equations be solved for P_t. Solving the second equation for P_t yields

$P_t = {1 \over \beta_2} Q_t - {\alpha_2 \over \beta_2} - {\gamma_2 \over \beta_2} F_t - {\gamma_3 \over \beta_2} Y_t$

You can estimate the intercepts of a system of simultaneous equations by applying PROC CALIS on the uncorrected covariance (UCOV) matrix of the data set that is augmented by an additional constant variable with the value 1. In the following example, the uncorrected covariance matrix is augmented by an additional variable INTERCEPT by using the AUGMENT option. The PROC CALIS statement contains the options UCOV and AUG to compute and analyze an augmented UCOV matrix from the input data set FOOD.

   Data food;                                                 
   Title 'Food example of KMENTA(1971, p.565 & 582)';         
     Input Q P D F Y;                                         
     Label  Q='Food Consumption per Head'                     
            P='Ratio of Food Prices to General Price'         
            D='Disposable Income in Constant Prices'          
            F='Ratio of Preceding Years Prices'               
            Y='Time in Years 1922-1941';                      
     datalines;                                               
     98.485  100.323   87.4   98.0   1                        
     99.187  104.264   97.6   99.1   2                        
    102.163  103.435   96.7   99.1   3                        
    101.504  104.506   98.2   98.1   4                        
    104.240   98.001   99.8  110.8   5                        
    103.243   99.456  100.5  108.2   6                        
    103.993  101.066  103.2  105.6   7                        
     99.900  104.763  107.8  109.8   8                        
    100.350   96.446   96.6  108.7   9                        
    102.820   91.228   88.9  100.6  10                        
     95.435   93.085   75.1   81.0  11                        
     92.424   98.801   76.9   68.6  12                        
     94.535  102.908   84.6   70.9  13                        
     98.757   98.756   90.6   81.4  14                        
    105.797   95.119  103.1  102.3  15                        
    100.225   98.451  105.1  105.0  16                        
    103.522   86.498   96.4  110.5  17                        
     99.929  104.016  104.4   92.5  18                        
    105.223  105.769  110.7   89.3  19                        
    106.232  113.490  127.1   93.0  20                        
    ;                                                         
                                                              
   proc calis ucov aug data=food pshort;                      
    Title2 'Compute ML Estimates With Intercept';             
    Lineqs                                                    
         Q = alf1 Intercept + alf2 P + alf3 D + E1,           
         P = gam1 Intercept + gam2 Q + gam3 F + gam4 Y + E2;  
    Std                                                       
         E1-E2 = eps1-eps2;                                   
    Cov                                                       
         E1-E2 = eps3;                                        
    Bounds                                                    
         eps1-eps2 >= 0. ;                                    
   run;

The following, essentially equivalent model definition uses program code to reparameterize the model in terms of the original equations; the output is displayed in Output 17.2.1.

   proc calis data=food ucov aug pshort;                                   
      Lineqs                                                               
            Q = alphal Intercept + beta1 P + gamma1 D + E1,                
            P = alpha2_b Intercept + gamma2_b F + gamma3_b Y + _b Q + E2;  
      Std                                                                  
            E1-E2 = eps1-eps2;                                             
      Cov                                                                  
            E1-E2 = eps3;                                                  
                                                                           
      Parameters alpha2 (50.) beta2 gamma2 gamma3 (3*.25);                 
      alpha2_b = -alpha2 / beta2;                                          
      gamma2_b = -gamma2 / beta2;                                          
      gamma3_b = -gamma3 / beta2;                                          
      _b       = 1 / beta2;                                                
                                                                           
     Bounds                                                                
            eps1-eps2 >= 0. ;                                              
   RUN;

Output 17.2.1: Food Example of Kmenta

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Pattern and Initial Values

LINEQS Model Statement
		Matrix	Rows	Columns	Matrix Type
Term 1	1	_SEL_	6	8	SELECTION
	2	_BETA_	8	8	EQSBETA	IMINUSINV
	3	_GAMMA_	8	6	EQSGAMMA
	4	_PHI_	6	6	SYMMETRIC

The 2 Endogenous Variables
Manifest	Q P
Latent

The 6 Exogenous Variables
Manifest	D F Y Intercept
Latent
Error	E1 E2

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Maximum Likelihood Estimation

Parameter Estimates	10
Functions (Observations)	21
Lower Bounds	2
Upper Bounds	0

Optimization Start
Active Constraints	0	Objective Function	2.3500065042
Max Abs Gradient Element	203.9741437	Radius	62167.829174

Iteration	Restarts	Function Calls	Active Constraints	Objective Function	Objective Function Change	Max Abs Gradient Element	Lambda	Ratio Between Actual and Predicted Change
1	0	2	0	1.19094	1.1591	3.9410	0	0.688
2	0	5	0	0.32678	0.8642	9.9864	0.00127	2.356
3	0	7	0	0.19108	0.1357	5.5100	0.00006	0.685
4	0	10	0	0.16682	0.0243	2.0513	0.00005	0.867
5	0	12	0	0.16288	0.00393	1.0570	0.00014	0.828
6	0	13	0	0.16132	0.00156	0.3643	0.00004	0.864
7	0	15	0	0.16077	0.000557	0.2176	0.00006	0.984
8	0	16	0	0.16052	0.000250	0.1819	0.00001	0.618
9	0	17	0	0.16032	0.000201	0.0662	0	0.971
10	0	18	0	0.16030	0.000011	0.0195	0	1.108
11	0	19	0	0.16030	6.116E-7	0.00763	0	1.389
12	0	20	0	0.16030	9.454E-8	0.00301	0	1.389
13	0	21	0	0.16030	1.461E-8	0.00118	0	1.388
14	0	22	0	0.16030	2.269E-9	0.000465	0	1.395
15	0	23	0	0.16030	3.59E-10	0.000182	0	1.427

Optimization Results
Iterations	15	Function Calls	24
Jacobian Calls	16	Active Constraints	0
Objective Function	0.1603035477	Max Abs Gradient Element	0.0001820805
Lambda	0	Actual Over Pred Change	1.4266532872
Radius	0.0010322573

GCONV convergence criterion satisfied.

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function	0.1603
Goodness of Fit Index (GFI)	0.9530
GFI Adjusted for Degrees of Freedom (AGFI)	0.0120
Root Mean Square Residual (RMR)	2.0653
Parsimonious GFI (Mulaik, 1989)	0.0635
Chi-Square	3.0458
Chi-Square DF	1
Pr > Chi-Square	0.0809
Independence Model Chi-Square	534.27
Independence Model Chi-Square DF	15
RMSEA Estimate	0.3281
RMSEA 90% Lower Confidence Limit	.
RMSEA 90% Upper Confidence Limit	0.7777
ECVI Estimate	1.8270
ECVI 90% Lower Confidence Limit	.
ECVI 90% Upper Confidence Limit	3.3493
Probability of Close Fit	0.0882
Bentler's Comparative Fit Index	0.9961
Normal Theory Reweighted LS Chi-Square	2.8142
Akaike's Information Criterion	1.0458
Bozdogan's (1987) CAIC	-0.9500
Schwarz's Bayesian Criterion	0.0500
McDonald's (1989) Centrality	0.9501
Bentler & Bonett's (1980) Non-normed Index	0.9409
Bentler & Bonett's (1980) NFI	0.9943
James, Mulaik, & Brett (1982) Parsimonious NFI	0.0663
Z-Test of Wilson & Hilferty (1931)	1.4250
Bollen (1986) Normed Index Rho1	0.9145
Bollen (1988) Non-normed Index Delta2	0.9962
Hoelter's (1983) Critical N	25

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Maximum Likelihood Estimation

Q	=	-0.2295	*	P	+	0.3100	*	D	+	93.6193	*	Intercept	+	1.0000		E1
				beta1				gamma1				alphal
P	=	4.2140	*	Q	+	-0.9305	*	F	+	-1.5579	*	Y	+	-218.9	*	Intercept	+	1.0000	E2
				_b				gamma2_b				gamma3_b				alpha2_b

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable	Parameter	Estimate
D		10154
F		9989
Y		151.05263
Intercept		1.05263
E1	eps1	3.51274
E2	eps2	105.06746

Covariances Among Exogenous Variables
Var1	Var2	Parameter	Estimate
D	F		9994
D	Y		1101
F	Y		1046
D	Intercept		102.66842
F	Intercept		101.71053
Y	Intercept		11.05263
E1	E2	eps3	-18.87270

Food example of KMENTA(1971, p.565 & 582)

The CALIS Procedure

Covariance Structure Analysis: Maximum Likelihood Estimation

Q	=	-0.2278	*	P	+	0.3016	*	D	+	0.9272	*	Intercept	+	0.0181		E1
				beta1				gamma1				alphal
P	=	4.2467	*	Q	+	-0.9048	*	F	+	-0.1863	*	Y	+	-2.1849	*	Intercept	+	0.0997	E2
				_b				gamma2_b				gamma3_b				alpha2_b

Squared Multiple Correlations
	Variable	Error Variance	Total Variance	R-Square
1	Q	3.51274	10730	0.9997
2	P	105.06746	10565	0.9901

Correlations Among Exogenous Variables
Var1	Var2	Parameter	Estimate
D	F		0.99237
D	Y		0.88903
F	Y		0.85184
D	Intercept		0.99308
F	Intercept		0.99188
Y	Intercept		0.87652
E1	E2	eps3	-0.98237

Additional PARMS and Dependent Parameters
The Number of Dependent Parameters is 4
Parameter	Estimate	Standard Error	t Value
alpha2	51.94453	.	.
beta2	0.23731	.	.
gamma2	0.22082	.	.
gamma3	0.36971	.	.
_b	4.21397	.	.
gamma2_b	-0.93053	.	.
gamma3_b	-1.55794	.	.
alpha2_b	-218.89288	.	.

You can obtain almost equivalent results by applying the SAS/ETS procedure SYSLIN on this problem.

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