HURST EXPONENT AND FINANCIAL MARKET PREDICTABILITY
Bo Qian Khaled Rasheed Department of Computer Science University of Georgia Athens, GA 30601 USA [qian, khaled]@cs.uga.edu

ABSTRACT The Hurst exponent (H) is a statistical measure used to classify time series. H=0.5 indicates a random series while H>0.5 indicates a trend reinforcing series. The larger the H value is, the stronger trend. In this paper we investigate the use of the Hurst exponent to classify series of financial data representing different periods of time. Experiments with backpropagation Neural Networks show that series with large Hurst exponent can be predicted more accurately than those series with H value close to 0.50. Thus Hurst exponent provides a measure for predictability. KEY WORDS Hurst exponent, time series analysis, neural networks, Monte Carlo simulation, forecasting

In time series forecasting, the first question we want to answer is whether the time series under study is predictable. If the time series is random, all methods are expected to fail. We want to identify and study those time series having at least some degree of predictability. We know that a time series with a large Hurst exponent has strong trend, thus it’s natural to believe that such time series are more predictable than those having a Hurst exponent close to 0.5. In this paper we use neural networks to test this hypothesis. Neural networks are nonparametric universal function approximators [9] that can learn from data without assumptions. Neural network forecasting models have been widely used in financial time series analysis during the last decade [10],[11],[12]. As universal function approximators, neural networks can be used for surrogate predictability. Under the same conditions, a time series with a smaller forecasting error than another is said to be more predictable. We study the Dow-Jones index daily return from Jan. 2, 1930 to May. 14, 2004 and calculate the Hurst exponent of each period of 1024 trading days. We select 30 periods with large Hurst exponents and 30 periods with Hurst exponents close to random series, and then we use these data to train neural networks. We compare forecasting errors for these two groups and find that the errors are significantly different. This research is done using Matlab. All Matlab programs generating result for this paper can be downloaded from www.arches.uga.edu/~qianbo/research. The remainder of the paper is organised as follows: Section 2 describes the Hurst exponent in detail. Section 3 then describes the monte carlo simulation process we used to generate data with similar structure to the financial series of interest to us. Section 4 describes a scramble test that we conducted to help verify that there is structure in the series due to the order of samples. Section 5 describes neural networks and their use to verify that sequences with larger values of the Hurst exponent can be more accurately learned and predicted than those with lower Hurst exponent values. Finally, the paper is concluded in section 6.

1. Introduction
The Hurst exponent, proposed by H. E. Hurst [1] for use in fractal analysis [2],[3], has been applied to many research fields. It has recently become popular in the finance community [4],[5],[6] largely due to Peters’ work [7],[8]. The Hurst exponent provides a measure for longterm memory and fractality of a time series. Since it is robust with few assumptions about underlying system, it has broad applicability for time series analysis. The values of the Hurst exponent range between 0 and 1. Based on the Hurst exponent value H, a time series can be classified into three categories. (1) H=0.5 indicates a random series. (2) 0 0.54 0.9863 0.9784 0.9902 0.9754 0.9773 0.9477 0.9265 0.9598 0.9658 0.9705 0.9641 0.9824 0.9557 0.9767 0.9793 0.9731 0.0162 H>0.65 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.93 0.9218 0.9256 0.9326 0.937 0.9376 0.9402 0.9445 0.9498 0.948 0.9486 0.9451 0.9468 0.9467 0.9542 0.55>H> 0.54 0.9747 0.9738 0.9635 0.957 0.9518 0.9542 0.9766 0.9901 0.9777 0.9814 0.9778 0.9968 0.9966 0.9977 0.9883

1 2 3 4 5 6 7 8 9 10 Mean Std.

Table 5.1. NRMSE for dimension 3 with hidden nodes 8, 9, 10, 11 and 12
Dimension 4, Hidden nodes 6 7 8 9 10 0.9572 0.9572 0.9557 0.9558 0.9633 0.9534 0.9554 0.9523 0.9518 0.9574 0.9373 0.9406 0.9414 0.9437 0.9404 0.9419 0.9471 0.9392 0.9332 0.9376 0.9691 0.9678 0.9597 0.9669 0.9662 0.9907 0.9939 0.9836 0.9948 0.9876 0.9902 0.9816 0.9872 0.9766 0.9855 0.9852 0.9842 0.9802 0.9865 0.9878 0.9809 0.9729 0.9722 0.9669 0.9741 0.9916 0.9957 0.9991 1.0025 0.9959 0.9698 0.9696 0.9671 0.9679 0.9696 0.0210 0.0193 0.0204 0.0225 0.0203 MIN 0.9557 0.9518 0.9373 0.9332 0.9597 0.9836 0.9766 0.9802 0.9669 0.9916 0.9637 0.0196 Std. 0.0031 0.0023 0.0023 0.0052 0.0037 0.0046 0.0053 0.0029 0.0050 0.0041 0.0038 0.0012

1 2 3 4 5 6 7 8 9 10 Mean Std.

Table 5.4. NRMSE for two groups We ran the unpaired student’s t test for the null hypothesis that the mean values of the two groups are equal. After calculation, the t statistic is 7.369 and p-value is 7.0290e010. This indicates the two means are significantly different and the chance of equality is essentially 0. This result confirms that the time series with larger Hurst exponent can be predicted more accurately.

Table 5.2. NRMSE for dimension 4 with hidden nodes 6, 7, 8, 9 and 10
Dimension 5, Hidden nodes 5 6 7 8 9 0.9578 0.9560 0.9617 0.9589 0.9622 0.9441 0.9466 0.9456 0.9456 0.9427 0.9410 0.9395 0.9449 0.9428 0.9396 0.9546 0.9453 0.9414 0.9409 0.9479 0.9659 0.9501 0.9671 0.9653 0.9653 0.9906 0.9919 0.9898 0.9901 0.9891 0.9803 0.9819 0.9805 0.9840 0.9837 0.9912 0.9980 1.0009 0.9991 1.0049 0.9770 0.9747 0.9742 0.9761 0.9689 0.9909 0.9984 0.9975 0.9977 0.9951 0.9693 0.9682 0.9704 0.9701 0.9699 0.0194 0.0233 0.0220 0.0226 0.0227 MIN 0.9560 0.9427 0.9395 0.9409 0.9501 0.9891 0.9803 0.9912 0.9689 0.9909 0.9650 0.0217 Std. 0.0026 0.0015 0.0023 0.0056 0.0071 0.0010 0.0017 0.0050 0.0032 0.0031 0.0033 0.0020

1 2 3 4 5 6 7 8 9 10 Mean Std.

6. Conclusion
In this paper, we analyze the Hurst exponent for all 1024trading-day periods of the Dow-Jones index from Jan.2, 1930 to May 14, 2004. We find that the periods with large Hurst exponents can be predicted more accurately than those with H values close to random series. This suggests that stock markets are not totally random in all periods. Some periods have strong trend structure and this structure can be learnt by neural networks to benefit forecasting.

Since the Hurst exponent provides a measure for predictability, we can use this value to guide data selection before forecasting. We can identify time series with large Hurst exponents before we try to build a model for prediction. Furthermore, we can focus on the periods with large Hurst exponents. This can save time and effort and lead to better forecasting.

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