Testing for Autocorrelation
The following options on the OLS command can be used
to obtain test statistics for detecting the presence of autocorrelation
in the residuals.
RSTAT |
Lists residual statistics including the Durbin-Watson statistic. |
DWPVALUE |
Computes the p-value for the Durbin-Watson test statistic. |
DLAG |
Computes Durbin's h statistic as a test for AR(1) errors when lagged dependent variables are included as regressors. The one-period lagged dependent variable must be listed as the first explanatory variable. |
Examples
Appendix
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Using the Durbin-Watson test
The Durbin-Watson test statistic is designed for detecting errors that follow a first-order autoregressive process. This statistic also fills an important role as a general test of model misspecification. See, for example, the discussion in Gujarati [1995, pp. 462-464].
The DWPVALUE option on the OLS
command computes a p-value for the Durbin-Watson test statistic.
Suppose the Durbin-Watson test statistic, d, has a calculated value of
DW. For a test of the null hypothesis of no autocorrelation in the
errors against the alternative hypothesis of positive autocorrelation
the p-value is:
p-value = P(d < DW)
The computation of a p-value is useful if the Durbin-Watson test statistic falls in the inconclusive range given in statistical tables. If the p-value is less than a selected level of significance (say 0.05) then there is evidence to reject the null hypothesis.
If the alternative hypothesis of interest is negative autocorrelation then the p-value is:
p-value = P(d > DW) = 1 - P(d < DW)
Following the OLS / DWPVALUE command the p-value
for the Durbin-Watson test is available in the temporary variable
$CDF. Therefore, when testing for negative autocorrelation,
a p-value can be computed with the commands:
OLS . . . / DWPVALUE GEN1 PVAL=1-$CDF PRINT PVAL |
Example
This example uses the Theil textile data set.
The SHAZAM commands
(filename: DW.SHA)
below first
estimate an equation with PRICE as the explanatory variable.
But economic theory suggests that INCOME is an important
variable in a demand equation. A statistical result is that if important
variables are omitted from the regression then the OLS estimator is biased.
The second OLS regression is the preferred model specification that
includes both PRICE and INCOME
as explanatory variables.
SAMPLE 1 17 READ (THEIL.txt) YEAR CONSUME INCOME PRICE OLS CONSUME PRICE / RSTAT DWPVALUE * Now include the variable INCOME in the regression equation OLS CONSUME INCOME PRICE / RSTAT DWPVALUE * Compute a p-value for testing for negative autocorrelation GEN1 PVAL=1-$CDF PRINT PVAL STOP |
The SHAZAM output can be inspected. The first OLS regression reports the results:
|
The estimation uses 17 observations and there are 2 estimated coefficients (including the intercept parameter). If we ignore the p-value and rely on tables printed at the end of textbooks we find that the lower and upper critical values are 1.133 and 1.381 (for a 5% significance level) and 0.874 and 1.102 (for a 1% significance level). When compared with the reported Durbin-Watson statistic the finding is that at a 5% level there is evidence for positive autocorrelation but at the 1% level the null hypothesis of no autocorrelation is not rejected. The computed p-value verifies this conclusion.
When the variable INCOME is added to the regression the
SHAZAM estimation results report:
|
By inspecting the p-value, the conclusion is that when both
PRICE and INCOME are included in the regression
there is no evidence to reject the null hypothesis of no
autocorrelation in the errors.
The regression equation that omitted INCOME showed
evidence for autocorrelated errors. However, this appears to reflect
that an important variable has been omitted - rather than a need
to correct for autocorrelation. That is, the omitted variable
INCOME is highly autocorrelated and when this variable
is included in the regression (as economic theory would typically suggest)
the autocorrelation in the residuals disappears.
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SHAZAM output with Durbin-Watson test statistics
|_SAMPLE 1 17
|_READ (THEIL.txt) YEAR CONSUME INCOME PRICE
UNIT 88 IS NOW ASSIGNED TO: THEIL.txt
4 VARIABLES AND 17 OBSERVATIONS STARTING AT OBS 1
|_OLS CONSUME PRICE / RSTAT DWPVALUE
OLS ESTIMATION
17 OBSERVATIONS DEPENDENT VARIABLE = CONSUME
...NOTE..SAMPLE RANGE SET TO: 1, 17
DURBIN-WATSON STATISTIC = 1.19071
DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE = 0.018346
NEGATIVE AUTOCORRELATION TEST P-VALUE = 0.981655
R-SQUARE = 0.8961 R-SQUARE ADJUSTED = 0.8892
VARIANCE OF THE ESTIMATE-SIGMA**2 = 61.594
STANDARD ERROR OF THE ESTIMATE-SIGMA = 7.8482
SUM OF SQUARED ERRORS-SSE= 923.91
MEAN OF DEPENDENT VARIABLE = 134.51
LOG OF THE LIKELIHOOD FUNCTION = -58.0829
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 15 DF P-VALUE CORR. COEFFICIENT AT MEANS
PRICE -1.3233 0.1163 -11.38 0.000-0.947 -0.9466 -0.7508
CONSTANT 235.49 9.079 25.94 0.000 0.989 0.0000 1.7508
DURBIN-WATSON = 1.1907 VON NEUMANN RATIO = 1.2651 RHO = 0.38554
RESIDUAL SUM = 0.00000 RESIDUAL VARIANCE = 61.594
SUM OF ABSOLUTE ERRORS= 102.14
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.8961
RUNS TEST: 6 RUNS, 9 POS, 0 ZERO, 8 NEG NORMAL STATISTIC = -1.7451
|_* Now include the variable INCOME in the regression equation
|_OLS CONSUME INCOME PRICE / RSTAT DWPVALUE
OLS ESTIMATION
17 OBSERVATIONS DEPENDENT VARIABLE = CONSUME
...NOTE..SAMPLE RANGE SET TO: 1, 17
DURBIN-WATSON STATISTIC = 2.01855
DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE = 0.301270
NEGATIVE AUTOCORRELATION TEST P-VALUE = 0.698730
R-SQUARE = 0.9513 R-SQUARE ADJUSTED = 0.9443
VARIANCE OF THE ESTIMATE-SIGMA**2 = 30.951
STANDARD ERROR OF THE ESTIMATE-SIGMA = 5.5634
SUM OF SQUARED ERRORS-SSE= 433.31
MEAN OF DEPENDENT VARIABLE = 134.51
LOG OF THE LIKELIHOOD FUNCTION = -51.6471
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 14 DF P-VALUE CORR. COEFFICIENT AT MEANS
INCOME 1.0617 0.2667 3.981 0.001 0.729 0.2387 0.8129
PRICE -1.3830 0.8381E-01 -16.50 0.000-0.975 -0.9893 -0.7846
CONSTANT 130.71 27.09 4.824 0.000 0.790 0.0000 0.9718
DURBIN-WATSON = 2.0185 VON NEUMANN RATIO = 2.1447 RHO = -0.18239
RESIDUAL SUM = -0.53291E-14 RESIDUAL VARIANCE = 30.951
SUM OF ABSOLUTE ERRORS= 72.787
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9513
RUNS TEST: 7 RUNS, 9 POS, 0 ZERO, 8 NEG NORMAL STATISTIC = -1.2423
|_* Compute a p-value for testing for negative autocorrelation
|_GEN1 PVAL=1-$CDF
..NOTE..CURRENT VALUE OF $CDF = 0.30127
|_PRINT PVAL
PVAL
0.6987301
|_STOP
|
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