Tests for serial correlation in regression analysis based on the periodogram of least-squares residuals

A well-known procedure for testing for serial correlation is to plot out the sample path of the cumulated periodogram and to compare the resulting graph with the Kolmogorov-Smirnov limits. The paper considers small-sample aspects of this procedure when the periodogram is calculated from the residuals from least-squares regression. It is shown that for a test against an excess of low-frequency relative to high-frequency variation in the errors of the regression model, a pair of lines can be drawn on the graph such that if the path crosses the upper line the hypothesis of serial independence is definitely rejected, while if the path fails to cross the lower line the hypothesis is definitely accepted. In the intermediate case the test is inconclusive. Similar procedures are given for tests against an excess of high-frequency variation and for two-sided tests. To facilitate the tests a table of significance values of the appropriate modified Kolmogorov-Smirnov statistics is given. A further test based on the mean of the ordinates of the cumulated periodogram is considered. It is shown that bounding significance values are easily obtainable from significance values of the mean of a uniform distribution.