SHAZAM Sets of Linear Equations

Sets of Linear Equations

An economic model may contain a number of linear equations. It may be realistic to expect that the equation errors will be correlated. A set of equations that has contemporaneous cross-equation error correlation is called a seemingly unrelated regression (SUR) system. At first look the equations seem unrelated. But the equations are related through the correlation in the errors.

A set of seemingly unrelated regression equations can be estimated with the general command format:

SYSTEM neq / options
OLS depvar indeps
. . .
OLS depvar indeps

where neq is the number of equations and options is a list of desired options. After the SYSTEM command there must be one OLS command for each equation in the system. Options must not be specified on the OLS commands. Options are specified on the SYSTEM command.

Linear parameter restrictions can be imposed with the general command format:

SYSTEM neq / RESTRICT options
OLS depvar indeps
. . .
OLS depvar indeps
RESTRICT equation
. . .
RESTRICT equation
END

The RESTRICT commands are specified as linear functions of the variables in the system. The variable names represent the coefficients.

Example

The example in this section considers some more variations on the analysis of the investment demand data set that was explored in the previous section on pooling cross-section time-series data.

The SHAZAM commands (filename: SURE.SHA) below estimate a system of investment demand equations. Following model estimation hypothesis testing is considered to test for the validity of cross-equation parameter restrictions. 

SAMPLE 1 20
READ(FIRM1.txt) YEAR IGM FGM CGM ICHR FCHR CCHR / SKIPLINES=1
READ(FIRM2.txt) YEAR IGE FGE CGE IWH FWH CWH / SKIPLINES=1
READ(FIRM3.txt) YEAR IUS FUS CUS  / SKIPLINES=1

* Unrestricted SUR Estimation
SYSTEM 5 / DN
OLS IGM FGM CGM 
OLS ICHR FCHR CCHR 
OLS IGE FGE CGE 
OLS IWH FWH CWH 
OLS IUS FUS CUS  

* Hypothesis Testing
TEST
  TEST FGM=FCHR
  TEST FGM=FGE
  TEST FGM=FWH
  TEST FGM=FUS
END
TEST
  TEST CGM=CCHR
  TEST CGM=CGE
  TEST CGM=CWH
  TEST CGM=CUS
END
TEST FGM=FCHR
TEST FGM=FGE
TEST FGM=FWH
TEST FGM=FUS
TEST FCHR=FGE
TEST FCHR=FWH
TEST FCHR=FUS
TEST FGE=FWH
TEST FGE=FUS
TEST FWH=FUS
TEST CGM=CCHR
TEST CGM=CGE
TEST CGM=CWH
TEST CGM=CUS
TEST CCHR=CGE
TEST CCHR=CWH
TEST CCHR=CUS
TEST CGE=CWH
TEST CGE=CUS
TEST CWH=CUS

* Test for equality of slope parameters for
* General Electric (GE) and Westinghouse (WH)
TEST
  TEST FGE=FWH
  TEST CGE=CWH
END

* Test for equality of slope parameters for
* General Motors (GM), Chrysler (CHR) and U.S. Steel (US)
TEST
  TEST FGM=FCHR
  TEST FGM=FUS
  TEST CGM=CCHR
  TEST CGM=CUS
END
STOP

The SHAZAM output can be viewed.

Parameter estimates (with standard errors in parentheses) from SUR estimation as reported in Greene [2000, Table 15.6, p. 619] are:

  GM CH GE WE US
intercept -162.364  
(89.459) 		0.504  
(11.513) 		-22.439  
(25.519) 		1.089  
( 6.259) 		85.423  
(111.877)
F 0.120  
( 0.022) 		0.070  
( 0.017) 		0.037  
( 0.012) 		0.057  
( 0.011) 		0.101  
( 0.055)
C 0.383  
( 0.033) 		0.309  
( 0.026) 		0.131  
( 0.022) 		0.042  
( 0.041) 		0.400  
( 0.128)

Note: GM = General Motors, CH = Chrysler, GE = General Electric, WE = Westinghouse and US = U.S. Steel.

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SHAZAM output 

|_SAMPLE 1 20
|_READ(FIRM1.txt) YEAR IGM FGM CGM ICHR FCHR CCHR / SKIPLINES=1
UNIT 88 IS NOW ASSIGNED TO: FIRM1.txt
   7 VARIABLES AND       20 OBSERVATIONS STARTING AT OBS       1

|_READ(FIRM2.txt) YEAR IGE FGE CGE IWH FWH CWH / SKIPLINES=1
UNIT 88 IS NOW ASSIGNED TO: FIRM2.txt
   7 VARIABLES AND       20 OBSERVATIONS STARTING AT OBS       1

|_READ(FIRM3.txt) YEAR IUS FUS CUS  / SKIPLINES=1
UNIT 88 IS NOW ASSIGNED TO: FIRM3.txt
   4 VARIABLES AND       20 OBSERVATIONS STARTING AT OBS       1

|_* Unrestricted SUR Estimation

|_SYSTEM 5 / DN
|_OLS IGM FGM CGM
|_OLS ICHR FCHR CCHR
|_OLS IGE FGE CGE
|_OLS IWH FWH CWH
|_OLS IUS FUS CUS

MULTIVARIATE REGRESSION--    5 EQUATIONS
 10 RIGHT-HAND SIDE VARIABLES IN SYSTEM
MAX ITERATIONS =       1         CONVERGENCE TOLERANCE =   0.10000E-02
       20 OBSERVATIONS
DN OPTION IN EFFECT - DIVISOR IS N

ITERATION     0 COEFFICIENTS
  0.11928      0.37144      0.77948E-01  0.31572      0.26551E-01  0.15169
  0.52894E-01  0.92406E-01  0.15657      0.42387

ITERATION    0 SIGMA
   7160.3
  -282.76       149.87
   607.53      -21.376       660.83
   126.18       13.307       176.45       88.662
  -2222.1       418.08       904.95       546.19       8896.4

BREUSCH-PAGAN LM TEST FOR DIAGONAL COVARIANCE MATRIX
   CHI-SQUARE =    29.060     WITH   10 DEGREES OF FREEDOM

LOG OF DETERMINANT OF SIGMA=   32.163
LOG OF LIKELIHOOD FUNCTION =  -463.522

ITERATION    1 SIGMA INVERSE
  0.19714E-03
  0.14141E-03  0.82319E-02
 -0.15017E-03  0.87747E-03  0.35267E-02
 -0.57804E-03 -0.75210E-03 -0.68215E-02  0.35682E-01
  0.93359E-04 -0.39461E-03 -0.18686E-04 -0.16058E-02  0.25476E-03

ITERATION     1 COEFFICIENTS
  0.12049      0.38275      0.69546E-01  0.30854      0.37291E-01  0.13078
  0.57009E-01  0.41506E-01  0.10148      0.39999

ITERATION    1 SIGMA
   7216.0
  -313.70       152.85
   605.34       2.0474       700.46
   129.89       16.661       200.32       94.912
  -2686.5       455.09       1224.4       652.72       9188.2

LOG OF DETERMINANT OF SIGMA=   31.755
LOG OF LIKELIHOOD FUNCTION =  -459.440

SYSTEM R-SQUARE =  0.9805 ... CHI-SQUARE =   78.738     WITH  10 D.F.
LIKELIHOOD RATIO TEST OF DIAGONAL COVARIANCE MATRIX =   44.065     WITH  10 D.F.

VARIABLE       COEFFICIENT    ST.ERROR       T-RATIO
FGM         0.12049        0.21629E-01     5.5709
CGM         0.38275        0.32768E-01     11.680
FCHR        0.69546E-01    0.16898E-01     4.1157
CCHR        0.30854        0.25864E-01     11.930
FGE         0.37291E-01    0.12263E-01     3.0409
CGE         0.13078        0.22050E-01     5.9313
FWH         0.57009E-01    0.11362E-01     5.0174
CWH         0.41506E-01    0.41202E-01     1.0074
FUS         0.10148        0.54784E-01     1.8523
CUS         0.39999        0.12779         3.1300

 EQUATION  1 OF  5 EQUATIONS
DEPENDENT VARIABLE = IGM                  20 OBSERVATIONS

 R-SQUARE =   0.9207
VARIANCE OF THE ESTIMATE-SIGMA**2 =   7216.0
STANDARD ERROR OF THE ESTIMATE-SIGMA =   84.947
SUM OF SQUARED ERRORS-SSE=  0.14432E+06
MEAN OF DEPENDENT VARIABLE =   608.02
LOG OF THE LIKELIHOOD FUNCTION = -459.440

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
FGM       0.12049     0.2163E-01   5.571     0.000 0.804     0.3520     0.8589
CGM       0.38275     0.3277E-01   11.68     0.000 0.943     0.7791     0.4082
CONSTANT  -162.36      89.47      -1.815     0.070-0.403     0.0000    -0.2670

 EQUATION  2 OF  5 EQUATIONS
DEPENDENT VARIABLE = ICHR                 20 OBSERVATIONS

 R-SQUARE =   0.9119
VARIANCE OF THE ESTIMATE-SIGMA**2 =   152.85
STANDARD ERROR OF THE ESTIMATE-SIGMA =   12.363
SUM OF SQUARED ERRORS-SSE=   3057.0
MEAN OF DEPENDENT VARIABLE =   86.124
LOG OF THE LIKELIHOOD FUNCTION = -459.440

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
FCHR      0.69546E-01 0.1690E-01   4.116     0.000 0.706     0.2614     0.5598
CCHR      0.30854     0.2586E-01   11.93     0.000 0.945     0.8040     0.4344
CONSTANT  0.50430      11.52      0.4378E-01 0.965 0.011     0.0000     0.0059

 EQUATION  3 OF  5 EQUATIONS
DEPENDENT VARIABLE = IGE                  20 OBSERVATIONS

 R-SQUARE =   0.6876
VARIANCE OF THE ESTIMATE-SIGMA**2 =   700.46
STANDARD ERROR OF THE ESTIMATE-SIGMA =   26.466
SUM OF SQUARED ERRORS-SSE=   14009.
MEAN OF DEPENDENT VARIABLE =   102.29
LOG OF THE LIKELIHOOD FUNCTION = -459.440

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
FGE       0.37291E-01 0.1226E-01   3.041     0.002 0.594     0.3176     0.7077
CGE       0.13078     0.2205E-01   5.931     0.000 0.821     0.6746     0.5116
CONSTANT  -22.439      25.56     -0.8780     0.380-0.208     0.0000    -0.2194

 EQUATION  4 OF  5 EQUATIONS
DEPENDENT VARIABLE = IWH                  20 OBSERVATIONS

 R-SQUARE =   0.7264
VARIANCE OF THE ESTIMATE-SIGMA**2 =   94.912
STANDARD ERROR OF THE ESTIMATE-SIGMA =   9.7423
SUM OF SQUARED ERRORS-SSE=   1898.2
MEAN OF DEPENDENT VARIABLE =   42.892
LOG OF THE LIKELIHOOD FUNCTION = -459.440

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
FWH       0.57009E-01 0.1136E-01   5.017     0.000 0.773     0.6634     0.8917
CWH       0.41506E-01 0.4120E-01   1.007     0.314 0.237     0.1352     0.0829
CONSTANT   1.0889      6.284      0.1733     0.862 0.042     0.0000     0.0254

 EQUATION  5 OF  5 EQUATIONS
DEPENDENT VARIABLE = IUS                  20 OBSERVATIONS

 R-SQUARE =   0.4220
VARIANCE OF THE ESTIMATE-SIGMA**2 =   9188.2
STANDARD ERROR OF THE ESTIMATE-SIGMA =   95.855
SUM OF SQUARED ERRORS-SSE=  0.18376E+06
MEAN OF DEPENDENT VARIABLE =   405.46
LOG OF THE LIKELIHOOD FUNCTION = -459.440

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
FUS       0.10148     0.5478E-01   1.852     0.064 0.410     0.2362     0.4935
CUS       0.39999     0.1278       3.130     0.002 0.605     0.4732     0.2958
CONSTANT   85.423      111.9      0.7631     0.445 0.182     0.0000     0.2107

|_* Hypothesis Testing
|_TEST
|_  TEST FGM=FCHR
|_  TEST FGM=FGE
|_  TEST FGM=FWH
|_  TEST FGM=FUS
|_END
WALD CHI-SQUARE STATISTIC =   18.886206     WITH    4 D.F.  P-VALUE= 0.00083
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.21179
|_TEST
|_  TEST CGM=CCHR
|_  TEST CGM=CGE
|_  TEST CGM=CWH
|_  TEST CGM=CUS
|_END
WALD CHI-SQUARE STATISTIC =   106.54776     WITH    4 D.F.  P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.03754
|_TEST FGM=FCHR
TEST VALUE =  0.50947E-01 STD. ERROR OF TEST VALUE  0.29181E-01
ASYMPTOTIC NORMAL STATISTIC =   1.7459342      P-VALUE= 0.08082
WALD CHI-SQUARE STATISTIC =   3.0482863     WITH    1 D.F.  P-VALUE= 0.08082
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.32805
|_TEST FGM=FGE
TEST VALUE =  0.83202E-01 STD. ERROR OF TEST VALUE  0.22161E-01
ASYMPTOTIC NORMAL STATISTIC =   3.7543680      P-VALUE= 0.00017
WALD CHI-SQUARE STATISTIC =   14.095279     WITH    1 D.F.  P-VALUE= 0.00017
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.07095
|_TEST FGM=FWH
TEST VALUE =  0.63484E-01 STD. ERROR OF TEST VALUE  0.22245E-01
ASYMPTOTIC NORMAL STATISTIC =   2.8537966      P-VALUE= 0.00432
WALD CHI-SQUARE STATISTIC =   8.1441551     WITH    1 D.F.  P-VALUE= 0.00432
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.12279
|_TEST FGM=FUS
TEST VALUE =  0.19015E-01 STD. ERROR OF TEST VALUE  0.62293E-01
ASYMPTOTIC NORMAL STATISTIC =  0.30524615      P-VALUE= 0.76018
WALD CHI-SQUARE STATISTIC =  0.93175210E-01 WITH    1 D.F.  P-VALUE= 0.76018
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST FCHR=FGE
TEST VALUE =  0.32254E-01 STD. ERROR OF TEST VALUE  0.21753E-01
ASYMPTOTIC NORMAL STATISTIC =   1.4827680      P-VALUE= 0.13814
WALD CHI-SQUARE STATISTIC =   2.1986011     WITH    1 D.F.  P-VALUE= 0.13814
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.45483
|_TEST FCHR=FWH
TEST VALUE =  0.12536E-01 STD. ERROR OF TEST VALUE  0.20543E-01
ASYMPTOTIC NORMAL STATISTIC =  0.61024150      P-VALUE= 0.54170
WALD CHI-SQUARE STATISTIC =  0.37239468     WITH    1 D.F.  P-VALUE= 0.54170
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST FCHR=FUS
TEST VALUE = -0.31933E-01 STD. ERROR OF TEST VALUE  0.55319E-01
ASYMPTOTIC NORMAL STATISTIC = -0.57724010      P-VALUE= 0.56378
WALD CHI-SQUARE STATISTIC =  0.33320613     WITH    1 D.F.  P-VALUE= 0.56378
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST FGE=FWH
TEST VALUE = -0.19718E-01 STD. ERROR OF TEST VALUE  0.10235E-01
ASYMPTOTIC NORMAL STATISTIC =  -1.9265070      P-VALUE= 0.05404
WALD CHI-SQUARE STATISTIC =   3.7114292     WITH    1 D.F.  P-VALUE= 0.05404
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.26944
|_TEST FGE=FUS
TEST VALUE = -0.64187E-01 STD. ERROR OF TEST VALUE  0.53675E-01
ASYMPTOTIC NORMAL STATISTIC =  -1.1958498      P-VALUE= 0.23176
WALD CHI-SQUARE STATISTIC =   1.4300568     WITH    1 D.F.  P-VALUE= 0.23176
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.69927
|_TEST FWH=FUS
TEST VALUE = -0.44469E-01 STD. ERROR OF TEST VALUE  0.51806E-01
ASYMPTOTIC NORMAL STATISTIC = -0.85837692      P-VALUE= 0.39068
WALD CHI-SQUARE STATISTIC =  0.73681094     WITH    1 D.F.  P-VALUE= 0.39068
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST CGM=CCHR
TEST VALUE =  0.74202E-01 STD. ERROR OF TEST VALUE  0.46049E-01
ASYMPTOTIC NORMAL STATISTIC =   1.6113636      P-VALUE= 0.10710
WALD CHI-SQUARE STATISTIC =   2.5964927     WITH    1 D.F.  P-VALUE= 0.10710
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.38513
|_TEST CGM=CGE
TEST VALUE =  0.25196     STD. ERROR OF TEST VALUE  0.34111E-01
ASYMPTOTIC NORMAL STATISTIC =   7.3865591      P-VALUE= 0.00000
WALD CHI-SQUARE STATISTIC =   54.561255     WITH    1 D.F.  P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.01833
|_TEST CGM=CWH
TEST VALUE =  0.34124     STD. ERROR OF TEST VALUE  0.47802E-01
ASYMPTOTIC NORMAL STATISTIC =   7.1386421      P-VALUE= 0.00000
WALD CHI-SQUARE STATISTIC =   50.960211     WITH    1 D.F.  P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.01962
|_TEST CGM=CUS
TEST VALUE = -0.17245E-01 STD. ERROR OF TEST VALUE  0.13956
ASYMPTOTIC NORMAL STATISTIC = -0.12356820      P-VALUE= 0.90166
WALD CHI-SQUARE STATISTIC =  0.15269100E-01 WITH    1 D.F.  P-VALUE= 0.90166
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST CCHR=CGE
TEST VALUE =  0.17776     STD. ERROR OF TEST VALUE  0.35586E-01
ASYMPTOTIC NORMAL STATISTIC =   4.9953052      P-VALUE= 0.00000
WALD CHI-SQUARE STATISTIC =   24.953074     WITH    1 D.F.  P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.04008
|_TEST CCHR=CWH
TEST VALUE =  0.26704     STD. ERROR OF TEST VALUE  0.48302E-01
ASYMPTOTIC NORMAL STATISTIC =   5.5284573      P-VALUE= 0.00000
WALD CHI-SQUARE STATISTIC =   30.563840     WITH    1 D.F.  P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.03272
|_TEST CCHR=CUS
TEST VALUE = -0.91447E-01 STD. ERROR OF TEST VALUE  0.12223
ASYMPTOTIC NORMAL STATISTIC = -0.74817908      P-VALUE= 0.45435
WALD CHI-SQUARE STATISTIC =  0.55977194     WITH    1 D.F.  P-VALUE= 0.45435
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_TEST CGE=CWH
TEST VALUE =  0.89277E-01 STD. ERROR OF TEST VALUE  0.35511E-01
ASYMPTOTIC NORMAL STATISTIC =   2.5140381      P-VALUE= 0.01194
WALD CHI-SQUARE STATISTIC =   6.3203873     WITH    1 D.F.  P-VALUE= 0.01194
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.15822
|_TEST CGE=CUS
TEST VALUE = -0.26921     STD. ERROR OF TEST VALUE  0.12354
ASYMPTOTIC NORMAL STATISTIC =  -2.1790789      P-VALUE= 0.02933
WALD CHI-SQUARE STATISTIC =   4.7483850     WITH    1 D.F.  P-VALUE= 0.02933
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.21060
|_TEST CWH=CUS
TEST VALUE = -0.35848     STD. ERROR OF TEST VALUE  0.12323
ASYMPTOTIC NORMAL STATISTIC =  -2.9091345      P-VALUE= 0.00362
WALD CHI-SQUARE STATISTIC =   8.4630637     WITH    1 D.F.  P-VALUE= 0.00362
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.11816
|_* Test for equality of slope parameters for
|_* General Electric (GE) and Westinghouse (WH)
|_TEST
|_  TEST FGE=FWH
|_  TEST CGE=CWH
|_END
WALD CHI-SQUARE STATISTIC =   7.7527986     WITH    2 D.F.  P-VALUE= 0.02073
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.25797
|_* Test for equality of slope parameters for
|_* General Motors (GM), Chrysler (CHR) and U.S. Steel (US)
|_TEST
|_  TEST FGM=FCHR
|_  TEST FGM=FUS
|_  TEST CGM=CCHR
|_  TEST CGM=CUS
|_END
WALD CHI-SQUARE STATISTIC =   10.641944     WITH    4 D.F.  P-VALUE= 0.03090
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.37587
|_STOP
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