Can linear regression analysis really predict the future?
- Thread starter tradrejoe
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Left axis is the avg length of the absolute distance traveled by each type of walk. It is plotted vs. number of steps on x axis. You can look up the random walk theoretical equation for this as sqrt(2n/pi); there is plenty of information on the web and statistics books.
Green Line -- is theoretical curve predicted by the equation above
Blue Line -- is an arbitrary random walk generated by GBM.
Orange Line - is curve generated by true market data.
You might need to use a constant scaling factor to align the plots, if you wish to reproduce, as distance itself is a relative term. It's the envelope shape you are more interested in.
P.S. As Maestro mentioned on his last post, I will likely not be discussing the ideas in this thread much longer as the exchange doesn't seem to be very bi-directional. Nevertheless, I truly hope someone gains something useful from the discourse exchanged thus far.
OK, giant leap into the unknown here, if I assume that my graphs are not in error, it seems that on larger timescales the relationship produced is linear with respect to time. A classical random walk has has a square root proportionality to time (or steps) as given by sqt(2 * n / pi). In my scant reading about Quantum Random Walks, it seems one of their properties is a linear proportionality to time (O(t) as opposed to O(sqt(t))), could this be what I am seeing in the graphs?
What I mean is that no matter which value you assign to t, it is continuous, and can be an infinite number of possibilities as you progress through time. No matter if its O(t) or O(sqrt(t))) the progression as time passes is based on the continuum and contributes to the randomness of any dataset. Across smaller and smaller intervals, prices specifically as with any function will look more and more linear, because the distance between the two gets shorter and shorter. With any 3 dimensional space projected into 4 dimensions through the passage of time, or if you're only looking at price and have a kind of 2 d curve, the movements are smaller and become more linearly proportional. As I said, it's not because what you are observing is proportionally linear, than it is that you are observing time as a continuum, be it on the x axis, or some other part of a progressive 3 dimensional object moving through time.
As it is used here, think about the range of any interval price chart, wouldn't you agree that as you parse smaller and smaller invetervals together, that prices become closer together and decrease in the range they may take as you go smaller. If you looked at a tick chart, that is as small an interval as it gets, and these prices I admit do appear linear, but as you increase the interval, the range of the bar widens, because time is a bigger factor. What you see at the tick level, is a much different story than what you see at a daily level. Daily goes to weekly, weekly to monthly, annually, 10 year, 15 year, max, the ranges will widen not just from the open, but from the high, low, and close.
When you use linear regression, there is a point at a small enough interval that you can keep regressing past prices onto the dataset, and can work just as well with physics. What I'm saying that data does appear linearly proportional as you go to smaller intervals.