Using Durbin's h test
When the regression includes lagged dependent variables the Durbin-Watson d-statistic is not valid as a test for autocorrelated residuals. The d-statistic tends to be biased towards 2. In this case, Durbin's h test can be considered.
This example returns to the analysis of the data set on monthly sales and advertising expenditures of a dietary weight control product. An application with this data set that illustrated regression with a lagged dependent variable was given in a previous section.
The next set of SHAZAM commands show the use of the DLAG
option on the OLS command to obtain Durbin's h test
as a test for AR(1) errors when lagged dependent variables are included
as regressors. The DLAG option requires that the
1-period lagged dependent variable is the first
explanatory variable in the list of variables.
Warning: In some cases the h-statistic cannot be calculated. If this occurs then no test statistic is reported on the SHAZAM output.
SAMPLE 1 36 READ (SALES.txt) SALES ADVERT GENR L1SALES=LAG(SALES) * Adjust the sample period SAMPLE 2 36 OLS SALES L1SALES ADVERT / DLAG STOP |
As an alternative to the above commands a special lag notation can
be used. On the OLS command explanatory
variables can have the general form:
var(first.last)
where var is an explanatory variable and first and last specify the range of periods to use for lags. SHAZAM automatically deletes the required number of initial undefined observations. The use of the special lag notation is shown in the next list of SHAZAM commands.
SAMPLE 1 36 READ (SALES.txt) SALES ADVERT OLS SALES SALES(1.1) ADVERT / DLAG STOP |
The SHAZAM output can be viewed. The estimation results report:
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -0.41847
The term ASYMPTOTIC NORMAL refers to the distribution of
the test statistic. For "large samples" the test statistic
has a standard normal distribution. Therefore, for a test of the null
hypothesis of no autocorrelation against the 2-sided alternative of
autocorrelated errors, at a 5% level, the decision rule is
if -1.96 < h < 1.96
By applying this decision rule it can be seen that there is no evidence for autocorrelation in the residuals. As a cautionary note users should consider that this test may not be accurate in "small samples".
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SHAZAM output
|_SAMPLE 1 36
|_READ (SALES.txt) SALES ADVERT
UNIT 88 IS NOW ASSIGNED TO: SALES.txt
2 VARIABLES AND 36 OBSERVATIONS STARTING AT OBS 1
|_OLS SALES SALES(1.1) ADVERT / DLAG
OLS ESTIMATION
35 OBSERVATIONS DEPENDENT VARIABLE = SALES
...NOTE..SAMPLE RANGE SET TO: 2, 36
R-SQUARE = 0.6720 R-SQUARE ADJUSTED = 0.6515
VARIANCE OF THE ESTIMATE-SIGMA**2 = 12.142
STANDARD ERROR OF THE ESTIMATE-SIGMA = 3.4845
SUM OF SQUARED ERRORS-SSE= 388.53
MEAN OF DEPENDENT VARIABLE = 24.606
LOG OF THE LIKELIHOOD FUNCTION = -91.7859
VARIABLE SUM OF LAG COEFS STD ERROR T-RATIO MEAN LAG
SALES 0.52793 0.10212 5.1698 1.0000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 32 DF P-VALUE CORR. COEFFICIENT AT MEANS
SALES 0.52793 0.1021 5.170 0.000 0.675 0.5478 0.5252
ADVERT 0.14647 0.3308E-01 4.428 0.000 0.616 0.4692 0.1721
CONSTANT 7.4469 2.470 3.015 0.005 0.470 0.0000 0.3027
DURBIN-WATSON = 2.0275 VON NEUMANN RATIO = 2.0871 RHO = -0.05637
RESIDUAL SUM = 0.18874E-12 RESIDUAL VARIANCE = 12.142
SUM OF ABSOLUTE ERRORS= 99.086
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6720
RUNS TEST: 20 RUNS, 19 POS, 0 ZERO, 16 NEG NORMAL STATISTIC = 0.5631
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -0.41847
|_STOP
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