In section 3.3 of Kutner's Applied Statistical Linear Models:
Nonindependence of Error Terms
Whenever data are obtained in a time sequence or some other type of sequence, such as for adjacent geographic areas, it is a good idea to prepare a sequence plot of the residuals.
... When the error terms are independent, we expect the residuals in a sequence plot to fluctuate in a more or less random pattern around the base line 0. ...
In section 3.4
Tests for Randomness
A runs test is frequently used to test for lack of randomness in the residuals arranged in time order. Another test, specifically designed for lack of randomness in least squares residuals, is the Durbin-Watson test. This test is discussed in Chapter 12.
What is the difference of testing independence of error terms and testing of randomness in the residuals? I guess they are the same thing?