- The tests for auto-correlation are concerned with the
dependence between numbers in a sequence.
- The list of the 30 numbers on page 311 appears to have the
effect that every 5th number has a very large value. If this is a
regular pattern, we can't really say the sequence is random.
- The test computes the auto-correlation between every
*m*numbers (*m*is also known as the lag) starting with the*i*th number.Thus the autocorrelation between the following numbers would be of interest.

The value

*M*is the largest integer such that where*N*is the total number of values in the sequence.E.g. N = 17, i = 3, m = 4, then the above sequence would be 3, 7, 11, 15 (M = 2). The reason we require

*M+1*instead of*M*is that we need to have at least two numbers to test (M = 0) the autocorrelation. - Since a non-zero autocorrelation implies a lack of
independence, the following test is appropriate
- For large values of
*M*, the distribution of the estimator , denoted as , is approximately normal if the values are uncorrelated. - Form the test statistic

which is distributed normally with a mean of zero and a variance of one. - The actual formula for
and the standard
deviation is

and

- After computing , do not reject the null hypothesis of
independence if

where is the level of significance. - See Example 8.12 on page 312.