SHAZAM Durbin-Watson test for Fixed Effects Models

Computing a p-value for the Durbin-Watson test for Fixed Effects Models

A discussion of the generalization of the Durbin-Watson test to testing for serial correlation in the OLS residuals from the fixed effects model is given in Bhargava, Franzini and Narendranathan [1982].

The Durbin-Watson test statistic can be expressed as:

      d = ê'Aê / ê'ê

where ê are the OLS residuals and the A matrix is suitably constructed for the panel data. The exact distribution of d is calculated using the Imhof routine that computes the cumulative distribution function for a quadratic form in normal variables. This is implemented with the DISTRIB command in SHAZAM.

The SHAZAM commands (filename: FIRMDW.SHA) below do the computations. 

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')'

GEN1 NC=5
GEN1 NT=20    
* Create cross-section dummy variables.
MATRIX CSDUM=SEAS(100,-NC)

* OLS estimation with dummy variables
SAMPLE 1 100
POOL I F C CSDUM / NOCONSTANT NCROSS=5 OLS HETCOV RSTAT RESID=E

* Compute a p-value for the Durbin-Watson statistic
GEN1 K=$K
GEN1 NOBS=100
MATRIX X=(F|C|CSDUM)
MATRIX M=IDEN(NOBS)-X*INV(X'X)*X'
* Generate the A matrix
SAMPLE 1 NT
GENR D=2
IF(TIME(0).EQ.1) D=1
IF(TIME(0).EQ.NT) D=1
MATRIX A=IDEN(NT,2)
MATRIX A=-(A+A')+DIAG(D)
MATRIX ASTAR=IDEN(NC)@A
* Get the eigenvalues of MA (sorted in descending order)
MATRIX MA=M*ASTAR
MATRIX EDW=EIGVAL(MA)
* Determine the number of non-zero eigenvalues.
GEN1 DF=NOBS-K
* Compute the DW statistic
MATRIX DWTEST=(E'ASTAR*E)/(E'E)
* Compute the p-value with the IMHOF routine.
SAMPLE 1 1
DISTRIB DWTEST / EIGENVAL=EDW TYPE=IMHOF NEIGEN=DF CDF=PVALUE
PRINT DWTEST DF PVALUE
STOP

The SHAZAM output can be viewed.

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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')'

|_GEN1 NC=5
|_GEN1 NT=20
|_* Create cross-section dummy variables.
|_MATRIX CSDUM=SEAS(100,-NC)

|_* OLS estimation with dummy variables
|_SAMPLE 1 100
|_POOL I F C CSDUM / NOCONSTANT NCROSS=5 OLS HETCOV RSTAT RESID=E

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
CSDUM     -76.067      74.99      -1.014     0.313-0.105    -0.1142    -0.0611
CSDUM     -29.374      11.93      -2.462     0.016-0.247    -0.0441    -0.0236
CSDUM     -242.17      35.36      -6.849     0.000-0.579    -0.3635    -0.1945
CSDUM     -57.899      12.77      -4.534     0.000-0.425    -0.0869    -0.0465
CSDUM      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

|_* Compute a p-value for the Durbin-Watson statistic
|_GEN1 K=$K
..NOTE..CURRENT VALUE OF $K   =   7.0000
|_GEN1 NOBS=100
|_MATRIX X=(F|C|CSDUM)
|_MATRIX M=IDEN(NOBS)-X*INV(X'X)*X'
|_* Generate the A matrix
|_SAMPLE 1 NT
|_GENR D=2
|_IF(TIME(0).EQ.1) D=1
|_IF(TIME(0).EQ.NT) D=1
|_MATRIX A=IDEN(NT,2)
|_MATRIX A=-(A+A')+DIAG(D)
|_MATRIX ASTAR=IDEN(NC)@A
|_* Get the eigenvalues of MA (sorted in descending order)
|_MATRIX MA=M*ASTAR
|_MATRIX EDW=EIGVAL(MA)
|_* Determine the number of non-zero eigenvalues.
|_GEN1 DF=NOBS-K
|_* Compute the DW statistic
|_MATRIX DWTEST=(E'ASTAR*E)/(E'E)
|_* Compute the p-value with the IMHOF routine.
|_SAMPLE 1 1
|_DISTRIB DWTEST / EIGENVAL=EDW TYPE=IMHOF NEIGEN=DF CDF=PVALUE

                DATA         CDF      1-CDF
  DWTEST
 ROW     1    0.77452     0.28982E-09  1.0000

|_PRINT DWTEST DF PVALUE
      DWTEST         DF             PVALUE
  0.7745177       93.00000      0.2898198E-09
|_STOP
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