Run Test

A run in a sequence of symbols is a group of consecutive symbols of one kind preceded and followed by symbols of another kind. For example, in the sequence, the runs can be exhibited by putting vertical bars or uninterrupted bars at the changes of the symbol (also known as ups and downs). In this sequence we have 241 runs of ups and 292 runs of downs.

We then consider the series of weekly changes in S&P 500 index and we calculate its median. The median in this runs test was 59. To each one of the excess returns we assign the symbol ‘+’ if it is above the median and ‘-’ if it is below. The run test is based on the intuitive notion that an unusually large or an unusually small value of r would suggest lack of randomness. In this sequence, r is even number as 14.

A second practice we adopt by using the 533 weekly S&P Index run test with the same length as the excess returns time series. The series are consisted of random numbers belonging in the interval [0,1]. We plot the distribution of r for these 533 weekly S&P Index run test and then again we compare the runs we found in the S&P Index with this distribution.

Moreover we convert all the distributions to standard normal distributions N (0, 1). The conversion is done according to: Z =X - E(r)/var(r), where X is the values of the initial distributions. In this run test shows that Z = -21.94.Then we calculate the probability of the value of Z, which in case is 1.09. Thus, from the S&P 500 Index weekly run test we found that approximately 45% of the random time it had upward trends and approximately 55% times downwards. To sum up the results we obtained from the run test, we did not find evidence against the randomness of the S&P weekly Index.