Alternate Runs Test in Time Series

For example, suppose that 15 data points are available, and the +/— value 

Looking at Table I (lower-tail and positive correlation), find i, the number of positive e,s and the number of negative e,s. Using a lower-tail test (i.e., positive correlation, because there are few runs), = 10 and 2 = 5. However, looking at Table I, we see that rii must be less than 2 so we simply exchange them = 5 and 2 = 10. There are four runs, or r = 4. The probability of this pattern being random is 0.029. There is a good indication of positive correlation. 

When larger sample sizes are used, and 2 10, a normal approximation can be made  

A lower-tail test approximation (positive correlation) can be completed using the normal Z tables (Table A), where 

If too many runs are present (negative correlation), the same formula is used, but — is used to compensate for an upper-tail test 

---