Detecting multifractal stochastic processes under heavy-tailed effects
Introduction
The importance of scaling relations in financial data was first stressed in the work of B.B. Mandelbrot. Early references are the seminal papers [1], [2]; see also [3]. A first concept of the scaling relation was self-affinity (or self-similarity, see [3, Chapter E6] for explanation of difference). Later the notion “monofractal” has also been used. As a generalization models allowing a richer form of scaling were introduced by Yaglom [4] and later called multifractal in the work of Frisch and Parisi [5]. Multifractals have been introduced as measures to model turbulence. The concept can be easily generalized to stochastic processes, thus extending the notion of self-similar stochastic processes.
In a series of papers [6], [7], [8], [9], the authors develop a theory of multifractal stochastic processes and a new model for financial time series, called the Multifractal model of asset returns (MMAR) (the name Brownian motion in multifractal time (BMMT) is used by Mandelbrot later). BMMT is constructed by compounding a standard (or fractional) Brownian motion with a random time process, which is specified to be multifractal. It incorporates most of the broadly accepted properties of financial data, such as long range dependence, volatility clustering and heavy tails. The multifractal property in this model is built using the notion of multiplicative cascade. Later, many models have been built possessing the multiscaling property, see e.g. [10], [11], [12], [13], [14].
Although multifractal models are very appealing, there is certain controversy over its use. Many authors have reported to find no evidence of multifractal scaling in different data sets and report spurious multiscaling for different model types (see [15], [16], [17], [18], [19], [20], [21], [22]). On the other hand, there is a range of papers confirming the multifractal behavior in various contexts by different methods: [23], [24], [25], [26], [27], [28] and, of course [7], [9].
In order to detect the multifractal property of a certain data set, one needs statistical methods. For multifractal stochastic process the multiscaling property is usually defined in terms of the moment scaling. This gives a simple detection method based on estimating the scaling function using a partition function. While for self-similar processes the scaling function is linear, for multifractal it should be nonlinear but always concave. Thus by estimating the scaling function, it is possible to distinguish the scaling nature of the process. Another detection method is based on the estimation of the multifractal spectrum. The spectrum can be obtained as a Legendre transform of the estimated scaling function provided so-called multifractal formalism holds (see [29]). However if the scaling function is unreliable, then the same is true for the spectrum.
In this paper we want to stress out that concavity of the estimated scaling function can be attributed to the presence of heavy tails in the data rather than multifractality. We derive an asymptotic form of the estimated scaling function for a large class of processes with stationary, heavy-tailed and weakly dependent increments. Estimation will yield a bilinear scaling function when the tail index is less than 2. This result is known for -stable Lévy processes (see [24], [16]). Processes we consider, unlike stable Lévy motion, are not assumed to be self-similar or to satisfy the moment scaling relation (see Eq. (3) below). Our results also show that this class of processes will behave as if they obey the moment scaling relation. When the tail index is larger than 2, scaling function will have a shape that is hard to recognize as bilinear. We illustrate through examples that this shape can be mistakenly regarded as evidence of multifractality. Therefore estimated scaling functions can be misleading, especially for financial data which is widely believed to be heavy-tailed (see, e.g., [30]).
Some authors define multifractality in terms of wavelets. This is usually done by basing the definition of the partition function on wavelet decomposition of the process (see e.g. [29], [31]). This leads to different methods for multifractal analysis based on wavelets. However, this type of definition is also sensitive to diverging moments. This has been noted in [32], where a wavelet based estimator of the tail index is proposed.
In the next section we recall some facts related to multifractal processes and statistical methods for analyzing multifractality using the partition function. In Section 3 we establish the asymptotic behavior of the partition function for a certain class of stochastic processes, in particular for processes with stationary independent heavy-tailed increments. Using this we derive an asymptotic form of the scaling function and multifractal spectrum for these processes. Results show that nonlinearity of the scaling function and a non-trivial spectrum can be caused by the presence of heavy tails. In Section 4, we present examples that indicate that empirical facts considered typical for multifractals can be reproduced by a simple heavy-tailed Lévy process.
Section snippets
Multifractal stochastic processes
The best known scaling relation is self-similarity. A stochastic process is said to be self-similar if for some and for any where equality is in finite dimensional distributions. The exponent H is usually called the Hurst parameter or index and we say is H-ss. Brownian motion is known to be self-similar with exponent and an -stable Lévy process is -s.s. Both of these have stationary and independent increments. On the other hand, fractional Brownian
Asymptotic behavior of the estimated scaling function
In this section we analyze the behavior of the estimated scaling function when the underlying process has heavy-tailed increments. A part of this analysis under different assumptions has also been made in [20].
Simulations and examples
In this section we provide examples showing that nonlinearity of the estimated scaling functions can be reconstructed just by using a process with heavy tailed increments. For this purpose we set to be a Student Lévy process, i.e. a stochastic process with stationary independent increments such that and has Student’s t-distribution.
It is important to stress that we do not advocate using Student Lévy process as a model in any of the examples. Besides independence of increments
Summary and discussion
We provided a rigorous proof that estimating the scaling function using the partition function can lead to nonlinear estimates under the presence of heavy tails. These results shed a new light on many data sets that have been claimed to be multifractal by using the partition function method. This is particularly important for financial data which can produce nonlinear scaling functions due to its heavy-tailed properties. Scaling functions can be estimated correctly but only when the range of
Proof of Theorem 1
For the sake of completeness we provide a short version of the proof of Theorem 1. Full proof can be found in [50]. We split the proof into two parts depending on whether or . The case follows from this due to monotonicity of in q. For simplicity we assume here that are symmetric around 0. Notation follows the one from Section 3.
(a) Suppose first that . For , we first show the upper bound on the limit:
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