# Fractal market hypothesis testing

I would like to do an analysis on the AEX stock exchange index for the last 20 years, but I ran into some issues. It would be really appreciated if you can answer my questions:

1. In order to apply tecniques like the fractal dimension, R/S analysis, the Hurst exponent, V-analysis, which would be the series taken into consideration? (the original price series or the log-return series)? I read a lot of articles on this subject and there is not a consistent approach (most of them I saw that ar applied on the log-return series, but,in my case, when I tried, for instance, to calculate the fractal dimension in R on both the original and the log-returns series, in the first case the results were as expected, whereas for the log-returns the values were above 2- which is absurd for a financial series)
2. For the analysis shall I take into consideration the entire series at once or split it into periods (before crisis, during crisis and after crisis)? Moreover, shall I use moving windows for the analysis? Are the functions in R consistent enough?
3. Which are the best methods to determine the H exponent?
4. For predicting the time series on short term using the chaos theory, what method shall I apply and on which series? I heard that the MSM – Markov-switcing multifractal method is a good alternative of the ARCH family models. Is it true? (it is calculated based on the return series).
5. Are there any other methods from the chaos theory that can help me predict the evolution of the AEX index? What analysis based on local memory, short memory and long memory can I make? Thank you in advance!

I suggest you read my book, Fractal Market Analysis, if you wish to perform R/S Analysis. You should use log differences of the prices. If you use one subperiod you are calculating the local Hurst exponent. If you want to try to find the global hurst exponent you need to calculate the average r/s for contiguous increments and average them and see how they scale as the window expands through a log/log plot.

I’ve found that those who find values of two are usually using raw prices rather than returns of log differences of the prices. It’s a common mistake.

• Welcome to Quant.stackexchange Mr. Peters! – Alex C May 9 '18 at 15:41

So let us look closely at the series to analyze it.

For starters let us look at a daily series and ignore things like weekends and holidays and ignore things like rebalancing and weightings for now. We are going to pretend the AEX actually exists as a coherent concept.

In the raw form, we can think of the time series as $$x_{t+1}=\beta{x}_t+\epsilon_{t+1},\beta>1.$$ Economic theory is based on the idea that all parameters are known, therefore this is a Brownian motion with drift conditional on $\Pr(\beta=k)=1$. If that is not true, then this is a Levy flight and no non-Bayesian solution exists to estimate $\beta$. Now if you take the logs, then the likelihood function for $\beta$ is the hyperbolic secant distribution, which has a finite mean and a finite, non-zero variance. It does not have a covariance matrix in its multivariate form, though. Keep that in mind because in rebalancing, for the process to be stationary, you are going to need to substitute firms with exactly the same log-mean and log-variance or you really have many time series with structural breaks four times per year. Since that is an event of measure zero, you will have a lot of work.

The Hurst Exponent is undefined for the raw data and is biased for the log data materially. The fractal dimension can be done on either, but there are mapping issues created by the log transformation. Real world data is limited to losses of -100%, truncating the sampling distribution for $\hat{\beta}-\beta.$ This shifts the median from being collocated with the mode which is the asymptotic location of $\beta$. This means that the logarithmic form shifts the true location to the median of the logarithmic data, which is also the mean. So it overstates the true center and understates the true scale, materially.

The memory of this process is infinite. It never forgets. It cannot forget as it is mathematically impossible to forget. The error terms may get lost in the jumble of many error terms that follow, but perfect memory is guaranteed. Consider the case of $x_0=0$ with a unit shock at time one where $\beta=1.1.$ If no further shocks happened, the unit shock would become $1.1$ at time two and $1.21$ at time three and so forth and would go to infinity as time went to infinity.

I can save you time here. I have proven that no computable, admissible, unbiased non-Bayesian estimator exists. Without going very formal, it is easy to see yourself. You can pick up a good argument regarding this in Jaynes tome on probability theory.

The likelihood function for $\beta$ isn't in the exponential family of distributions so no sufficient statistic exists. Since no point estimator contains all of the information in the sample, including for this like the Hurst exponent, any estimator will be noisy compared to a sufficient estimator. The Bayesian likelihood function is always a sufficient estimator, however, it produces an entire distribution of solutions rather than a point. This may not be useful depending on your real goal.

Further, truncation biases (badly) the estimate of the location so even if the Frequentist estimator had the same variability as the Bayesian, the Bayesian estimator would automatically stochastically dominate it. I am currently doing simulations on this problem and for the most recent sample of 1000 time series, each of length 10,000 the relative efficiency of the local estimators of the Bayesian versus the Frequentist is 16:1. A sixteen to one difference in short term efficiency is enormous. Moreso, the Bayesian estimator can never produce a parameter estimate that is impossible, while the Frequentist one must do so for some samples.

If you need a point estimate, then the proper solution is to calculate the Bayesian posterior predictive density and overlay a cost function. Minimizing the cost of being wrong guarantees the point estimate contains all of the information.

This is due to the fact that the Bayesian predictive distribution is $\Pr(x_{t+\Delta{t}}|x_0\dots{x}_t).$ Note that the prediction does not depend upon the parameters and so all the information regarding the parameters has been put into the distribution of the predictions. You are also now working in the sample space again from the parameter space that Bayesian methods work in. You can then use the distribution to make estimates down to a single point.

Of course, there are other solutions. The set of parameter estimates are minimally sufficient for inference and in the non-truncated case, the estimate of the pivotal value is normally distributed. It isn't clear if the pivot for the raw, but truncated data, is still valid.

You should not use window methods because there is permanent memory. You cannot use ARMA because this series has no average, there is no first moment estimate for this series. Only the zeroth moment exists.

You should drop fractal methods for the simple Bayesian predictive distribution. Don't forget though to update the AEX parameters at each rebalancing interval. Also remember you have no first or higher moment, so be prepared to think about this quite differently than usual. You will need to update the likelihood for weekends and holidays.