SHAZAM Testing for Autocorrelation

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:

DURBIN-WATSON STATISTIC  =   1.19071
DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE =    0.018346
              NEGATIVE AUTOCORRELATION TEST P-VALUE =    0.981655

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:

DURBIN-WATSON STATISTIC  =   2.01855
DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE =    0.301270
              NEGATIVE AUTOCORRELATION TEST P-VALUE =    0.698730

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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