Testing for AutocorrelationThe following options on the
ExamplesAppendix[SHAZAM Guide home] Using the Durbin-Watson testThe 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 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 Following the
ExampleThis example uses the Theil textile data set.
The SHAZAM commands
(filename:
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
By inspecting the p-value, the conclusion is that when both
The regression equation that omitted
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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