I've done the overall daily return distribution analysis that you first suggested and here are the results:Quote from science_trader:
... the problem with your approach is that it's not practically usable.
One example. Today is up. Then, from your study you would say that tomorrow would tend to be up too, but actually not, because your study says that there is a bias that may last more than just 2 days. I don't know if you follow me on that one.
Conditional probabilities are far more efficient and practical to work with. Then you know your odds (based on the past of course) of seeing precise events, not only saying that "as a whole" the market satisfies a statistical test or not.
1,256 positive deviations from the median daily return
1,256 negative deviations from the median daily return
1 zero deviation (ignored)
1,348 runs
The runs-test analysis yields a z-score of 3.632.
So the chance of getting 1,348 runs in a random sequence of 1,256 positive deviations and 1,256 negative deviations is one out of 7,112.
The daily returns of the S&P 500 from 2 Jan 1998 to 31 Dec 2007 are not remotely iid (independent and identically distributed).