Tests for Auto-correlation

• 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 ith number.

  Thus the autocorrelation $\rho_{im}$ between the following numbers would be of interest.
  

  $$R_i, R_{i+m}, R_{i+2m}, ... R_{i+(M+1)m}$$
  
  

  The value M is the largest integer such that $i + (M+1)m \le N$ 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

  $H_0:$	$\rho_{im} = 0$
  $H_1:$	$\rho_{im} \ne 0$

• For large values of M, the distribution of the estimator $\rho_{im}$, denoted as $\hat{\rho_{im}}$, is approximately normal if the values $R_i, R_{i+m}, R_{i+2m}, ... R_{i+(M+1)m}$ are uncorrelated.

• Form the test statistic
  
  $$Z_0 = \frac{\hat{\rho_{im}}} { \sigma_{\hat{\rho_{im}}}}$$
  
  
  which is distributed normally with a mean of zero and a variance of one.

• The actual formula for $\hat{\rho_{im}}$ and the standard deviation is
  
  $$\hat{\rho_{im}} = \frac{1}{M+1} \left[ \sum_{k=0}^M R_{i+km} R_{(k+1)m} \right] - 0.25$$
  
  
  and
  
  $$\sigma_{\hat{\rho_{im}}} = \frac{\sqrt{13M+7}}{12(M+1)}$$
  
  

• After computing $Z_0$, do not reject the null hypothesis of independence if
  
  $$- z_{\alpha/2} \le Z_0 \le z_{\alpha/2}$$
  
  
  where $\alpha$ is the level of significance.

• See Example 8.12 on page 312.