Sets of Linear EquationsAn 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:
where Linear parameter restrictions can be imposed with the general command format:
The ExampleThe 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:
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:
Note: GM = General Motors, CH = Chrysler, GE = General Electric, WE = Westinghouse and US = U.S. Steel.
[SHAZAM Guide home] 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 [SHAZAM Guide home] |