Pooling by OLS with Panel-Corrected Standard Errors and Dummy VariablesThe time series observations for all the cross-section units can be pooled and the regression coefficients can be estimated by OLS. Cross-section differences can be recognized by allowing different intercepts. Cross-section dummy variables are included as regressors and the equation is estimated by OLS. This is known as a fixed effects model. On the
Panel Corrected Standard ErrorsIt may be realistic to expect different error variances for the different cross-sections. For example, Greene [2000, p. 594] notes that for a cross-country comparison there may be variation in the scales of the variables in the model. It may also be realistic to expect cross-section contemporaneous error correlation. For the investment demand data set Greene [2000, p. 599] observes: . . . we have two automobile producers, two major suppliers to the electric utility industry, and one major supplier to all four of the others, U.S. Steel. It is very likely that the macroeconomic factors that affect these firms affect all of them to varying degrees. For the auto industry, the fates of GM and Chrysler are obviously tied both to the economy as a whole and to factors that are specific to the two firms. As such, it would seem reasonable to allow correlation of the disturbances across firms. With cross-section heteroskedasticity the OLS standard errors will be inconsistent. A method for computing a heteroskedastic-consistent covariance matrix for pooled regression models is discussed in Beck and Katz [1995], Beck et al. [1993] and Greene [2000, p. 594]. The Beck and Katz covariance matrix estimate gives "Panel Corrected
Standard Errors" (PCSE). Note that the The panel corrected standard errors are obtained as the square roots of the diagonal elements of the matrix: cov(b) =
(X'X) where is an N x N matrix with the (i,j)th element estimated by: (
tT=1
êi,têj,t
) Lagrange Multiplier TestsThe SHAZAM output reports a Lagrange multiplier statistic for testing for cross-section heteroskedasticity as suggested by Greene [2000, p. 596]. The output also reports the Breusch-Pagan Lagrange multiplier statistic for a test for a diagonal covariance matrix (that is, no cross-secion correlation). See Greene [2000, Equation 15-14, p. 601]ExampleThe SHAZAM commands (filename:
On the first The above SHAZAM program shows a general method for generating cross-section dummy variables. The SHAZAM output can be viewed. Test statistics calculated from the pooled OLS residuals are:
The first test statistic exceeds the critical value. Therefore, the null hypothesis of homoskedasticity is rejected (see Greene [2000, p. 598]). The second test statistic is computed from the OLS residuals. Greene [2000, p. 601] suggests that there may be some limitations to the interpretation of this test statistic. The table below shows the pooled OLS estimates with a comparison of alternative standard errors. These results can be compared with Greene [2000, Example 15.2, p. 594 and Table 15.1, p. 598].
The standard errors in the PCSE column are the Beck-Katz standard errors (reported in Greene [2000, Example 15.2, p. 594]). The standard errors in the HET column are the standard errors that assume no cross-section correlation (reported in the "Correct" column of Greene [2000, Table 15.1, p. 598]). Now consider the estimation results for the fixed effects model.
A separate dummy variable is included for each firm and the
Following model estimation the Testing for autocorrelationIt is also of interest to test for autocorrelation within cross-section
units. The
iN=1
tT=2
(êit The value for
iN=1
tT=2
êitêi,t The estimation results
for OLS with dummy variables report the Durbin-Watson
statistic
[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 |_* Pooling by OLS |_POOL I F C / NCROSS=5 OLS DN POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I THE DN OPTION IS IN EFFECT POOLING BY OLS 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 R-SQUARE = 0.7789 VARIANCE OF THE ESTIMATE-SIGMA**2 = 15709. STANDARD ERROR OF THE ESTIMATE-SIGMA = 125.33 SUM OF SQUARED ERRORS-SSE= 0.15709E+07 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -624.993 ASYMPTOTIC VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR -------- P-VALUE CORR. COEFFICIENT AT MEANS F 0.10509 0.1121E-01 9.378 0.000 0.690 0.5574 0.8114 C 0.30537 0.4285E-01 7.126 0.000 0.586 0.4236 0.3815 CONSTANT -48.030 21.16 -2.270 0.023-0.225 0.0000 -0.1929 |_* Pooling by OLS with Panel-Corrected Covariance Matrix |_POOL I F C / NCROSS=5 OLS HETCOV POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I POOLING BY OLS USING PANEL-CORRECTED COVARIANCE MATRIX 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 R-SQUARE = 0.7789 VARIANCE OF THE ESTIMATE-SIGMA**2 = 16195. STANDARD ERROR OF THE ESTIMATE-SIGMA = 127.26 SUM OF SQUARED ERRORS-SSE= 0.15709E+07 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -624.993 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 97 DF P-VALUE CORR. COEFFICIENT AT MEANS F 0.10509 0.8318E-02 12.63 0.000 0.789 0.5574 0.8114 C 0.30537 0.3304E-01 9.242 0.000 0.684 0.4236 0.3815 CONSTANT -48.030 10.81 -4.441 0.000-0.411 0.0000 -0.1929 |_* Generate a cross-section index |_* Set the number of time periods |_GEN1 NT=20 |_GENR CSINDEX=SUM(SEAS(NT)) |_* Get standard errors with correction for heteroskedasiticity but |_* assume the restriction of no cross-section correlation. |_POOL I F C / NCROSS=5 OLS HETCOV CSINDEX=CSINDEX POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS UNBALANCED PANELS TIME-PERIODS 1 20 2 20 3 20 4 20 5 20 DEPENDENT VARIABLE = I POOLING BY OLS USING PANEL-CORRECTED COVARIANCE MATRIX R-SQUARE = 0.7789 VARIANCE OF THE ESTIMATE-SIGMA**2 = 16195. STANDARD ERROR OF THE ESTIMATE-SIGMA = 127.26 SUM OF SQUARED ERRORS-SSE= 0.15709E+07 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -624.993 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 97 DF P-VALUE CORR. COEFFICIENT AT MEANS F 0.10509 0.9063E-02 11.60 0.000 0.762 0.5574 0.8114 C 0.30537 0.4095E-01 7.458 0.000 0.604 0.4236 0.3815 CONSTANT -48.030 14.20 -3.382 0.001-0.325 0.0000 -0.1929 |_* Now get the White standard errors |_OLS I F C / HETCOV OLS ESTIMATION 100 OBSERVATIONS DEPENDENT VARIABLE= I ...NOTE..SAMPLE RANGE SET TO: 1, 100 USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = 0.7789 R-SQUARE ADJUSTED = 0.7743 VARIANCE OF THE ESTIMATE-SIGMA**2 = 16195. STANDARD ERROR OF THE ESTIMATE-SIGMA = 127.26 SUM OF SQUARED ERRORS-SSE= 0.15709E+07 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -624.993 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 97 DF P-VALUE CORR. COEFFICIENT AT MEANS F 0.10509 0.9146E-02 11.49 0.000 0.759 0.5574 0.8114 C 0.30537 0.5911E-01 5.166 0.000 0.465 0.4236 0.3815 CONSTANT -48.030 15.02 -3.198 0.002-0.309 0.0000 -0.1929 |_* Create cross-section dummy variables. |_* Set the number of cross-sections |_GEN1 NC=5 |_MATRIX CSDUM=SEAS(100,-NC) |_DO #=1,NC |_ GENR D#=CSDUM:# |_ENDO _DO #=1,NC ****** EXECUTION BEGINNING FOR DO LOOP # = 1 #_ GENR D1=CSDUM:1 #_ ENDO #_ GENR D2=CSDUM:2 #_ ENDO #_ GENR D3=CSDUM:3 #_ ENDO #_ GENR D4=CSDUM:4 #_ ENDO #_ GENR D5=CSDUM:5 #_ ENDO ****** EXECUTION FINISHED FOR DO LOOP #= 5 |_* OLS estimation with dummy variables |_POOL I F C D1-D5 / NOCONSTANT NCROSS=5 OLS HETCOV RSTAT POOLED CROSS-SECTION TIME-SERIES ESTIMATION 100 TOTAL OBSERVATIONS 5 CROSS-SECTIONS 20 TIME-PERIODS DEPENDENT VARIABLE = I POOLING BY OLS USING PANEL-CORRECTED COVARIANCE MATRIX LM TEST FOR CROSS-SECTION HETEROSKEDASTICITY 33.468 CHI-SQUARE WITH 4 D.F. P-VALUE= 0.00000 BREUSCH-PAGAN LM TEST FOR DIAGONAL COVARIANCE MATRIX 28.322 CHI-SQUARE WITH 10 D.F. P-VALUE= 0.00160 R-SQUARE = 0.9375 VARIANCE OF THE ESTIMATE-SIGMA**2 = 4777.3 STANDARD ERROR OF THE ESTIMATE-SIGMA = 69.118 SUM OF SQUARED ERRORS-SSE= 0.44429E+06 MEAN OF DEPENDENT VARIABLE = 248.96 LOG OF THE LIKELIHOOD FUNCTION = -561.847 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 93 DF P-VALUE CORR. COEFFICIENT AT MEANS F 0.10598 0.1771E-01 5.985 0.000 0.527 0.5621 0.8183 C 0.34666 0.2716E-01 12.76 0.000 0.798 0.4808 0.4331 D1 -76.067 74.99 -1.014 0.313-0.105 -0.1142 -0.0611 D2 -29.374 11.93 -2.462 0.016-0.247 -0.0441 -0.0236 D3 -242.17 35.36 -6.849 0.000-0.579 -0.3635 -0.1945 D4 -57.899 12.77 -4.534 0.000-0.425 -0.0869 -0.0465 D5 92.539 39.35 2.352 0.021 0.237 0.1389 0.0743 DURBIN-WATSON = 0.7745 VON NEUMANN RATIO = 0.7823 RHO = 0.60606 RESIDUAL SUM = 0.41922E-11 RESIDUAL VARIANCE = 4777.3 SUM OF ABSOLUTE ERRORS= 4782.7 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9375 |_* Test for equality of firm intercepts |_TEST |_ TEST D1=D2 |_ TEST D1=D3 |_ TEST D1=D4 |_ TEST D1=D5 |_END F STATISTIC = 63.486725 WITH 4 AND 93 D.F. P-VALUE= 0.00000 WALD CHI-SQUARE STATISTIC = 253.94690 WITH 4 D.F. P-VALUE= 0.00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.01575 |_STOP [SHAZAM Guide home] |