SHAZAM Cross-Section Heteroskedasticity and Time-Wise Autoregression

Cross-Section Heteroskedasticity and Time-Wise Autoregression

In 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 POOL command in SHAZAM will estimate pooled time-series cross-section models under the assumptions of cross-section heteroskedasticity and first-order autoregressive errors. By default, SHAZAM assumes zero contemporaneous cross-section error correlation (cross-section independence). However, when T > N (where T is the number of time periods and N is the number of cross-sections) the FULL option can be used to allow for contemporaneous cross-section correlation as well as cross-section heteroskedasticity.

The SHAZAM commands (filename: FIRMGLS.SHA) below estimate the investment demand function by a number of alternative methods. 

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
* 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
* Assume cross-section correlation
POOL I F C / NCROSS=5 RHO=0 DN NOMULSIGSQ FULL

* 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 
STOP

The SHAZAM output can be viewed.

The first two model estimations assume serially uncorrelated errors with the RHO=0 option on the POOL command. The first estimation allows for cross-section heteroskedasticity. The second estimation uses the FULL option to recognize both cross-section heteroskedasticity as well as cross-section contemporaneous correlation. The DN and NOMULSIGSQ options are used to ensure that the calculation of the variance-covariance matrix of the parameter estimates conforms to the method used in Greene [2000].

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]).

Variable
name 		  Heteroskedasticity   Cross-section correlation
Estimate s.e. Estimate s.e.
F 0.0950 0.0074 0.0962 0.0055
C 0.3378 0.0302 0.3095 0.0180
intercept -36.2537 6.1244 -38.3613 5.3449

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.

Variable
name 		parameter
estimate 		standard
error 		t-statistic
F 0.0766 0.0082 9.3596
C 0.3794 0.0171 22.1775
    Firm Intercepts
General Motors 48.8186 51.5754 0.9465
Chrysler -11.9994 6.3580 -1.8873
General Electric -215.1412 49.2481 -4.3685
Westinghouse -47.6289 21.0275 -2.2651
U.S. Steel 138.1891 39.3775 3.5093

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.

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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

|_* 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
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