I first cited this table in my speech at the Smartlab conference in the spring of 2016 and repeated it at the 2018 conference, focusing on what I want to write below
What’s in the table? In the table of the share of sections RI (futures on the RTS index – my note) of 10 increments, both for individual periods and in general, which I referred to as “trends”. What did I consider “trend”? “Trend” I considered sections where the average of price increments (or increments of logarithms of prices, which is equivalent) is different from zero, and if it is greater than zero, then we attribute the segment to “trends up”, and if it is less than zero, to “trends down”.
What criterion was used? The usual modified Student’s t-test for the difference between increments of the logarithm (!) of the price from Gaussian process increments with mean zero and variance “almost equal” for 9 out of 10 trials (null hypothesis). Since we have a criterion for distinguishing a complex hypothesis from a simple one, the distribution of the statistics of the criterion is exactly known to us only with a simple hypothesis. And therefore, with a priori chosen boundaries of the criterion, we can only know the probabilities that a sequence of 10 values will fall into our “classes” if the null hypothesis is true.
I chose the boundaries so that, under the null hypothesis, the probability of falling into “trends up” (“trends down”) is 0.125. And we will be interested in the differences between the shares in real prices and the shares in the null hypothesis.
What does the alpha column from the table show us? Alpha is the probability that we will be wrong if we reject the null hypothesis. As we can see, this probability is close to zero for 8 periods out of 10 and in general for the population. BUT! If we look at the column where alpha1 stands, equal to the error probability of rejecting the hypothesis that the probability of “no trends” is 0.75=1-2*0.125, we will see that this error probability is very high, both for individual periods and for the aggregate as a whole.
What does it mean? And what with a probability greater than 0.99 price log increments cannot be Gaussian process increments with a constant(!) mean and “slowly varying” variance for the vast majority of the time.
And what is a Gaussian process with a constant (!) mean and a “slowly changing” variance for the vast majority of the time? it the Black-Scholes model and her ARCH-GARCH generalizations. It turns out that these models do not correspond to the market.
It is probably possible to come up with a model of a Gaussian process with a constant (!) Mean and variance “jumping like a sick squirrel”, which does not contradict the received frequencies, but the simplest model that explains such a frequency distribution is a model of a Gaussian process with a variable(!) mean .
So, for reference, the sample one-dimensional distribution of the frequencies of the values Gaussian process with variable(!) mean will be close to the generalized hyperbolic distribution, and not to the normal one, that is, it will have “heavy tails”. Therefore, the argument about “heavy tails” does not at all refute the hypothesis that the process is Gaussian.
Finishing with the analysis of the above table, we note one more interesting fact. Over the entire period under consideration, RI fell slightly (however, statistically this fall is indistinguishable from zero), and the share of “trends up” is greater than the share of “trends down”. What does it mean? Yes, a well-known fact: growth in the stock market is “slower”, and falls are “cooler”. And, by the way, in 8 out of 10 periods, the regularity “works”: if the share of “trends up” is greater than the share of “trends down”, then RI has grown (green lines) and vice versa (red lines). It seems logical, but there are exceptions – blue lines.
Let us add to the variability of the mean also the hypothesis of piecewise constancy. What does the average random walk curve look like, whose increments are such a Gaussian process?
This will be a broken line. Like this blue line if the amount has risen
Or so, if the amount fell
The green line in the first figure is the mean of a random walk with a constant positive mean.
Well, the trajectories (realizations) of our random walk will be curves that make “relatively small” oscillations around the blue line. “Relatively small” means much smaller than the average size of the rising and falling sections of the broken line.
And here we come to the most interesting. On such trajectories “backdating” it is easy to draw such figures from planimetry as triangles (they are also “pennants” or “butterflies”), parallelepipeds (they are also rectangles, “corridors” or “flags”), rhombuses (they are also “diamonds” ) and trapezoid. And if we neglect the error of a few percent (for daily periods) between the real local extremes of the trajectory and the Fibonacci levels, then it is possible to build an “Elliott wave”, and more than one with such errors.
BUT! If we additionally assume that the breaking points of the polyline are absolutely unpredictable, then the “value” of such drawings is zero. Why? Because it is impossible to predict breakpoints (see the assumption), and the best indicator that reveals these breakpoints after the fact is the simplest Bollinger Bands, built on trajectory increments (not values) from the previous breakpoint.
In contrast to the Gaussian and piecewise constancy of averages, it is impossible to statistically prove the hypothesis of absolute unpredictability of the breaking points of a broken line. It is possible to prove unpredictability only for particular cases of past information or to refute it in general. I will write about its proof for special cases of past information in the second part, but for now, anticipating the expected objection, I will make one important remark.
I foresee the objection: “What are the logarithms of prices – we work with prices, not with logarithms!”. Gentlemen, remember a high school course: the logarithm is an absolutely continuous monotonic function on the interval from 0 to infinity. What does it mean? And the fact that all the “antics and jumps” of the candlestick price chart through a one-to-one recalculation will “migrate” to the logarithm chart and vice versa. And along with this, any figures and patterns will “migrate”, as well as “Elliott waves” simply with other numerical values. This means that what I formulated in terms of the hypothesis of absolute unpredictability is also true for the prices themselves.