Cross-Section Heteroskedasticity and Time-Wise AutoregressionIn the previous example, inspection of the OLS estimated residuals found evidence for heteroskedasticity across firms and autocorrelation within firms. The generalized least squares (GLS) method makes use of this information by incorporating a general error covariance matrix in the estimation procedure. The estimation method is described in Parks [1967]. Further discussion is in Kmenta [1986, Section 12.2, pp. 616-625] and Greene [2000, Chapter 15]. The The SHAZAM commands
(filename:
The SHAZAM output can be viewed. The first two model estimations assume serially uncorrelated errors
with the The first two columns of the table below give the parameter estimates and standard errors obtained from the estimation with cross-section heteroskedasticity (see the FGLS column in Greene [2000, Table 15.1, p. 598]). The third and fourth columns give the parameter estimates and standard errors obtained from the estimation that incorporated both cross-section heteroskedasticity and contemporaneous correlation (see the FGLS row in Greene [2000, Table 15.2, p. 602]).
The final estimation results listed on the SHAZAM output recognize a different AR(1) error process for each firm. In addition, firm dummy variables are included to allow for differential firm intercepts. The results are summarized in the table below.
The above results reveal the variation in the firm intercepts. As a goodness of fit measure SHAZAM reports an R-square measure developed by Buse [1973]. Buse discusses that GLS can be viewed as a form of weighted least squares. A "generalized R-square" measure is appropriate for this situation.
[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 |_* Stack the data |_MATRIX I=(IGM'|ICHR'|IGE'|IWH'|IUS')' |_MATRIX F=(FGM'|FCHR'|FGE'|FWH'|FUS')' |_MATRIX C=(CGM'|CCHR'|CGE'|CWH'|CUS')' |_SAMPLE 1 100 |_* Assume that observations are uncorrelated across time. |_POOL I F C / NCROSS=5 RHO=0 DN NOMULSIGSQ POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I THE DN OPTION IS IN EFFECT MODEL ASSUMPTIONS: SAME FIXED RHO FOR EACH CROSS-SECTION= 0.0000 DIAGONAL PHI MATRIX OLS COEFFICIENTS 0.10509 0.30537 -48.030 LM TEST FOR CROSS-SECTION HETEROSKEDASTICITY 46.630 CHI-SQUARE WITH 4 D.F. P-VALUE= 0.00000 BREUSCH-PAGAN LM TEST FOR DIAGONAL COVARIANCE MATRIX 50.682 CHI-SQUARE WITH 10 D.F. P-VALUE= 0.00000 VARIANCES (DIAGONAL OF PHI MATRIX) 9410.9 755.85 34288. 633.42 33456. PHI MATRIX 9410.9 -168.05 755.85 -1916.0 -4163.3 34288. -1129.3 -80.382 2259.3 633.42 258.50 4035.9 -27898. -1170.7 33456. BUSE [1973] R-SQUARE = 0.9014 BUSE RAW-MOMENT R-SQUARE = 0.9326 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.94690 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.97309 SUM OF SQUARED ERRORS-SSE= 94.690 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -570.057 ASYMPTOTIC VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR -------- P-VALUE CORR. COEFFICIENT AT MEANS F 0.94991E-01 0.7409E-02 12.82 0.000 0.793 0.5038 0.7334 C 0.33781 0.3023E-01 11.18 0.000 0.750 0.4686 0.4221 CONSTANT -36.254 6.124 -5.920 0.000-0.515 0.0000 -0.1456 |_* Assume cross-section correlation |_POOL I F C / NCROSS=5 RHO=0 DN NOMULSIGSQ FULL POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I THE DN OPTION IS IN EFFECT MODEL ASSUMPTIONS: SAME FIXED RHO FOR EACH CROSS-SECTION= 0.0000 FULL PHI MATRIX - CROSS-SECTION CORRELATION OLS COEFFICIENTS 0.10509 0.30537 -48.030 LM TEST FOR CROSS-SECTION HETEROSKEDASTICITY 46.630 CHI-SQUARE WITH 4 D.F. P-VALUE= 0.00000 BREUSCH-PAGAN LM TEST FOR DIAGONAL COVARIANCE MATRIX 50.682 CHI-SQUARE WITH 10 D.F. P-VALUE= 0.00000 VARIANCES (DIAGONAL OF PHI MATRIX) 9410.9 755.85 34288. 633.42 33456. PHI MATRIX 9410.9 -168.05 755.85 -1916.0 -4163.3 34288. -1129.3 -80.382 2259.3 633.42 258.50 4035.9 -27898. -1170.7 33456. BUSE [1973] R-SQUARE = 0.9302 BUSE RAW-MOMENT R-SQUARE = 0.9662 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.96474 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.98221 SUM OF SQUARED ERRORS-SSE= 96.474 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -537.773 ASYMPTOTIC VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR -------- P-VALUE CORR. COEFFICIENT AT MEANS F 0.96189E-01 0.5475E-02 17.57 0.000 0.872 0.5102 0.7427 C 0.30953 0.1799E-01 17.21 0.000 0.868 0.4293 0.3868 CONSTANT -38.361 5.345 -7.177 0.000-0.589 0.0000 -0.1541 |_* Create cross-section dummy variables. |_* Set the number of cross-sections |_GEN1 NC=5 |_MATRIX CSDUM=SEAS(100,-NC) |_* Model with heteroskedasticity, cross-section correlation |_* and autocorrelation |_POOL I F C CSDUM / NCROSS=5 NOCONSTANT DN NOMULSIGSQ FULL POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I THE DN OPTION IS IN EFFECT MODEL ASSUMPTIONS: DIFFERENT ESTIMATED RHO FOR EACH CROSS-SECTION FULL PHI MATRIX - CROSS-SECTION CORRELATION OLS COEFFICIENTS 0.10598 0.34666 -76.067 -29.374 -242.17 -57.899 92.539 RHO VECTOR 0.62657 -0.47882E-01 0.85293 0.86680 0.51239 SAME ESTIMATED RHO FOR ALL CROSS-SECTIONS = 0.60606 VARIANCES (DIAGONAL OF PHI MATRIX) 4777.7 209.32 1525.5 258.00 7299.9 PHI MATRIX 4777.7 -397.48 209.32 529.74 247.89 1525.5 -102.05 116.82 542.72 258.00 -1330.0 343.71 473.97 489.59 7299.9 BUSE [1973] R-SQUARE = 0.9170 BUSE RAW-MOMENT R-SQUARE = 0.9720 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.96814 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.98394 SUM OF SQUARED ERRORS-SSE= 96.814 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -471.786 ASYMPTOTIC VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR -------- P-VALUE CORR. COEFFICIENT AT MEANS F 0.76618E-01 0.8186E-02 9.360 0.000 0.696 0.4064 0.5916 C 0.37943 0.1711E-01 22.18 0.000 0.917 0.5263 0.4741 CSDUM 48.819 51.58 0.9465 0.344 0.098 0.0733 0.0392 CSDUM -11.999 6.358 -1.887 0.059-0.192 -0.0180 -0.0096 CSDUM -215.14 49.25 -4.369 0.000-0.413 -0.3229 -0.1728 CSDUM -47.629 21.03 -2.265 0.024-0.229 -0.0715 -0.0383 CSDUM 138.19 39.38 3.509 0.000 0.342 0.2074 0.1110 |_STOP [SHAZAM Guide home] |