Durbin-Watson Test

Text: Section 13.9 (8th ed.), 14.9 (7th ed.)

$H_{o}$: No first order autocorrelation

$$\mbox{\bf Test Statistic: } d_{obs} = \frac{\displaystyl... ..._{t-1})^{2} } {\displaystyle \sum_{t=1}^{n} \hat{e}_{t}^{2} }$$

where $\hat{e}_{t}$ = residual.

NOTE: Text uses $\hat{R}_{t}$ instead of $\hat{e}_{t}$.

Possible alternatives:

1. $H_{a}$: Positive first order autocorrelation
   

   Reject $H_{o}$ if: $d_{obs} < d_{L, \alpha}$
   Don't reject $H_{o}$ if: $d_{obs} > d_{U, \alpha}$
   Unsure if: $d_{L, \alpha} < d_{obs} < d_{U, \alpha}$
   

2. $H_{a}$: Negative first order autocorrelation
   

   Reject $H_{o}$ if: $(4 - d_{obs}) < d_{L, \alpha}$
   Don't reject $H_{o}$ if: $(4 - d_{obs}) > d_{U, \alpha}$
   Unsure if: $d_{L, \alpha} < (4 - d_{obs}) < d_{U, \alpha}$
   

3. $H_{a}$: Positive or negative first order autocorrelation
   

   Reject $H_{o}$ if: $d_{obs} < d_{L, \alpha/2}$ or $(4 - d_{obs}) < d_{L, \alpha/2}$
   Don't reject $H_{o}$ if: $d_{obs} > d_{U, \alpha/2}$ or $(4 - d_{obs}) > d_{U, \alpha/2}$
   Unsure if: $d_{L, \alpha/2} < d_{obs} < d_{U, \alpha/2}$ and $d_{L, \alpha/2} < (4 - d_{obs}) < d_{U, \alpha/2}$
   

Table Values: Values of $d_{L, \alpha}$, $d_{U, \alpha}$ found in Table XIII, p. 1001-1002 (8th ed.), Table XV, p. 1032-1033 (7th ed.). In the table, $k$ is the number of explanatory variable terms in the regression equation, and $n$ is the number of observations.