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The CALIS Procedure

Example 19.2: 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 Pt. Solving the second equation for Pt 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 19.2.1.

   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 19.2.1: 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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