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Strategies & Market Trends : Waiting for the big Kahuna -- Ignore unavailable to you. Want to Upgrade?

To: Skeet Shipman who wrote (56762)	2/18/2002 1:40:07 AM
From: Moominoid	Read Replies (1) \| Respond to of 94695

A standard random walk has a Hurst exponent of 0.5. Its fractal dimension is 2. This means that in two dimensions the random walk eventually fills all the 2D space. A random walk is also said to be integrated of order 1. The changes in the random walk are stationary - have constant mean. If they were white noise as argued in finance theory, predicting stock prices from their past history would be impossible. It would be possible to predict stock prices a little if there was serial correlation in the changes in stock prices and there is plenty of this it turns out. An integrated random walk goes like this: yt = yt-1 + At + et At = At-1 + ut where et and ut are random errors so that A is a random walk and yt an integrated random walk with noise - this is called second order integration. You need to take the difference of the difference before you get a stationary process. This process has Hurst exponent of 1 and fractal dimension of 1 (a smooth line). If stocks were like this they would be very easy to forecast. It turns out as I think Vi told me (maybe someone else) that the Hurst exponent for stocks is between 0.6 and 0.7 which means that they have a level of integration between a random walk and an integrated random walk. This is called fractional integration. The effects of changes in prices decay slower than in a random walk with serially correlated noise but faster than the integrated random walk (where they never decay). Result is predictability to a degree, pseudo-cyclical behavior etc. Don't look for fixed length cycles - with the exception of 10 days they don't exist. The fractal dimension is between 1 and 2 - a fractal. This leads to the self similarity properties of stocks as in traditional TA and E-Wave. Over the weekend I have done some tests fractionally differencing the NDX. I don't have a formal test but it seems it is integrated of order about 1.25. What I can also see is that lags up to 50 periods back etc. are statistically significant. I have developed some simple forecasting models which work quite well too. I just backtested one from 6 March 2000. In theoretical unleveraged trading the account stays flat for about a year or so and then zooms to eight times value now. Seems at least 1000 observations are needed to get good results. Also a sign of fractional integration. Hope this makes some kind of sense! David