Self-similar processes

Let (X, 𝓐, P) be a probability space, with t ∈ [0, ∞) being time. Recall that X = {X(t, ω):t ≥ 0} is a random process or a random function from [0, ∞)×Ω to ℝ, if X(t, ω) is a random variable for all t ≥ 0 and all ω ∈ Ω, where ω belongs to a sample space Ω. Thus, we consider X as defining a sample function t ↦ X(t, ω) for all ω ∈ Ω. In this way, note that the points of Ω parametrize the functions X:[0, ∞) × Ω → ℝ, and P is a probability measure on this class of functions.

Let X(t, ω) and Y(t, ω) be two random functions. The notation X(t, ω) ∼ Y(t, ω) means that they have the same finite joint distribution functions. A random process X is said to be self-similar [71] if there exists a parameter H > 0 such that (1) for all a > 0 and t ≥ 0. Equivalently, X is also called a H-self-similar random process, and the parameter H is its self-similarity index or exponent. Thus, if X(t, ω) denotes space (with t being time), then Eq (1) establishes that every change of time scale a > 0 corresponds to a change of space scale a^H. Moreover, the bigger H, the more drastic is the change of the space variable. Additionally, Eq (1) implies also a scale-invariance of the finite dimensional distribution of the random process X.

A wide class of processes are self-similar. They include (fractional) Brownian motions as well as their generalizations, namely, (fractional) Lévy stable motions. In the fractional Brownian motion case, the self-similarity index H is also called Hurst exponent. Some authors also use the name Hurst exponent as a reference to the self-similarity index for any self-similar random process.

Generalized Hurst Exponent

The original procedure contributed by Hurst in [72] was generalized afterwards in [54, 73] to test for long-range correlations of time series. This approach is based on the qth-order moments of the distribution of the increments of the corresponding stochastic process X(t) which describes the evolution of the time series. According to that, the scaling properties of a (financial) time series are explored through a qth-order statistic we denote by K_q(τ), which is given as follows (see [74]): where T is the length of the time series X(t) (namely, the observation period) and τ ∈ {1, …, τ_max} is the number of days considered. Further, it is verified that the statistical properties of X(t) scale with the resolution of the time window.

Notice also that the next alternative expression for K_q(τ) could also be found in the literature (see [28, 69, 74]): where ⟨ ⋅ ⟩ denotes the sample average over the corresponding time window. Overall, the statistic K_q(τ) scales following a power-law like the next one: (2) Thus, the scaling behavior of K_q(τ) allows to find the so-called generalized Hurst exponent (GHE, herein) [75]. Observe that all the information about scaling properties of X(t) is contained in its scaling index H(q). This makes the GHE based analysis quite simple to test for scaling properties of time series.

As well as it happens with other tools to test for scaling properties of time series (see for example, MF-DFA [76], MF-DMA [63] and FD4 [77]), GHE provides a valid procedure to deal with multi-fractal (also multi-scaling) processes, namely, those ones in which the scaling exponent H(q) depends on each moment q. In such case, different exponents characterize the scaling properties of different qth-moments for the underlying distribution [78, 79]. According [28], the nonlinear properties of q H(q) vs. q (or equivalently, the linear patterns of H(q) vs. q) throw empirical evidence against the classical models proposed to fit actual (financial) time series, including (fractional) Brownian motions as well as (fractional) Lévy stable motions.

On the other hand, it is quite natural to apply GHE to properly study unifractal processes. Thus, scaling properties of such processes is uniquely determined through a single constant H which agrees with the self-similarity exponent (resp. Hurst exponent) of the time series. Note that in this case, the plot for q H(q) vs. q provides a straight line (equivalently, H(q) = H for each qth-order moment, that is, the self-similarity index remains constant).

In particular, if q = 1, then GHE provides a valid tool to analyze the scaling properties of the absolute deviations of the process under consideration. Thus, the quantity H(1) is expected to be close to the original Hurst exponent.

It is worth mentioning that in [74], it was shown that among all procedures considered to test for long-range correlation properties, only GHE (q = 1) and MF-DFA (q = 1) approaches could be used to properly calculate the self-similarity index for heavy-tailed distributions (see [74]). In addition to that, it was reported in such work that GHE (q = 1) performs much better than MF-DFA (q = 1). According to that, in this paper we will use GHE with q = 1 for Hurst exponent calculation purposes.

Note that it could be also possible to consider q = 2 in Eq (2) for long-range dependence detection purposes. Indeed, recall that in such case, K₂(τ) is related with the scaling of the autocorrelation function of the time series increments and it is also connected to the power spectrum. Indeed, such an autocorrelation function has the following expression: α(t, τ) = ⟨X(t+τ) X(t)⟩.

Hence, the empirical estimation of H(2) from K₂(τ) ∝ τ^{2 H(2)} is somehow equivalent to estimate the Hurst exponent (resp. the self-similarity exponent) H by means of any other classical procedure appeared in literature, such as R/S or MF-DFA.

We would like also to point out that GHE was revisited in [80] to test for scaling properties of financial time series. Further, though some recent works reported empirical evidence about multi-fractal behavior in financial time series, it seems that the source of multifractality has not been completely understood yet (see, for instance, [81, 82]).

The ETSP distribution

In this paper, we consider Elevated Two-Sided Power distributions. Our purpose is double: first, we will show that they allow to fit properly actual financial time series. Moreover, they also provide a suitable class of distributions valid to point out that previous information about the series under study should be taken into account before calculating its self-similarity exponent and concluding whether there exists memory in that series or not. According to that, in this section we will describe slightly ETSP distributions and we will also explain how to generate samples for such distributions.

ETSP distributions were first introduced in [83] as a new kind of heavy-tailed bounded distributions to model financial time series. The probability density function (pdf, herein) for an ETSP with support [0, 1] is given as the following parametric family of piecewise defined functions: (3) where Θ = {θ, m, n, λ, δ}. Note that the parameters in Θ provide different information about the distribution they describe. Indeed, θ ∈ [0, 1] is the threshold parameter, 0 ≤ λ, δ ≤ 1 are the elevation parameters, and m, n > 0 are the power parameters. Further, observe that the normalization constant 𝓒(Θ) may be described in terms of the parameters in Θ as follows: Fig 1 shows an ETSP pdf which properly fits the behavior of the log return of a real stock (HPQ in this case). Additionally, the cumulative distribution function (cdf for short) for Eq (3) has the following expression (see [83]): (4) where In particular, note that the class of TSP distributions (first introduced in [84]) remains as a particular case of the class of ETSP ones: indeed, it suffices with taking m = n and λ = δ = 0. In addition to that, a similar expression to Eq (3) for the description of the ETSP pdf (resp. to Eq (4) for the description of the ETSP cdf) could be obtained in the case of ETSP distributions whose support is a real interval [a, b]. Accordingly, note that in such case, Θ = {θ, m, n, λ, δ, a, b} would be the collection of parameters for this kind of generalized ETSP distributions and that $\frac{X-a}{b-a}$ would be an ETSP distribution in [0, 1].

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Fig 1. The figure above shows an ETSP pdf plotted over the interval [a = −0.19, b = 0.21] which properly fits the empirical pdf of a real stock corresponding to Hewlett-Packard Company (HPQ) from S&P500 index.

Note that in this case, its corresponding family of parameters was chosen to be as follows: Θ = {θ = 0.48, m = 13.6, n = 13.4, λ = 0.0185, δ = 0.0170}.

https://doi.org/10.1371/journal.pone.0127824.g001

Generating ETSP samples

In [83], it was shown how to calculate the quantile functions for an ETSP distribution. That procedure results especially useful for our purposes since it could be used to generate samples for this kind of distributions. In this paper, though, we will apply a faster (preliminary tests showed that this algorithm is about 10 times faster) and more accurate procedure instead of the former algorithm. Next, we describe how to artificially generate ETSP samples through this novel tecnhique. Indeed, let us consider F(θ) = G(x∣Θ) as in Eq (4). Hence, if π = F(θ), then the following algorithm could be used:

1. Let r₁, r₂, r₃ be three random (uniform) numbers in [0, 1].
2. If r₁ < π and r₂ < λ, then return θ*r₃.
3. If r₁ < π and r₂ ≥ λ, return $\theta *r_3^{\frac{1}{m}}$.
4. If r₁ ≥ π and r₂ < δ, return 1 − (1 − θ)*r₃.
5. If r₁ ≥ π and r₂ ≥ δ, return $1-(1-\theta )*r_3^{\frac{1}{n}}$.

Note that if we want to generate a sample from an ETSP distribution in an interval [a, b], we only have to apply the previous algorithm to generate a sample x of an ETSP in the interval [0, 1] and then apply the linear transformation a+(b − a) x.

Linking heavy-tailed distributions to self-similarity exponent

The self-similarity index of both fractional Brownian motions (FBMs onwards) and Lévy stable motions (LSMs, herein) has been widely analyzed in the literature (see, for example, [74]). Due to the Central Limit Theorem and its generalization to stable distributions (see [85]), if financial time series were sums of i.i.d. random variables, then the self-similarity index of such series should yield the type of stable distribution underlying the process. On the other hand, recall that in the FBM case, the self-similarity index is a “measure” to test for long-memory in the series.

In this study, we will focus only on daily changes series. Thus, note that we are not concerned with the intraday process which gives the daily changes, but on the distribution of the daily changes itself. For example, note that if the intraday process were a sum of normal distributed random variables with zero mean and exponentially distributed variances, then the corresponding daily changes would follow a Laplace distribution [86]. In addition to that, note that the Laplace distribution is also related with the TSP distribution (see [84, 87]). Hence, it would result pretty interesting to find a probability distribution function such that the sum of normal variables with zero mean and standard deviation following such unknown distribution could yield a TSP or an ETSP one as a result, though this problem still remains open.

Since both the TSP and the ETSP distributions provide a good fit for daily (log) returns of stock market prices, it is possible that the intraday process somehow could produce these kind of distributions or another type of similar ones, so it is quite interesting to investigate the self-similarity index of such processes, which constitutes the main goal of this paper.

Accordingly, in this section we study the self-similarity exponent of random processes modeled by means of heavy-tailed distributions with underlying ETSP distribution. A similar study carried out for Lévy stable processes can be found in [74].

Self-similarity index depends on the underlying distribution

Note that at a first glance, one could expect a similar result to the FBM case, namely, that a self-similarity index equal to 0.5 would imply that there is no memory in the series, but in fact, that is not the case. Actually, the results given by Monte Carlo simulation show that the self-similarity index (calculated by means of GHE approach) depends on the distribution, as well as it happens in the LSM case. Next, we analyze how the self-similarity exponent depends on the parameters of the ETSP distribution. To do this, we consider 1024-length time series. Overall, we can state that

• if the underlying distribution of the random process is a TSP (namely, an ETSP for which λ = δ = 0 and n = m), then H remains close to 0.5, whereas
• if the corresponding distribution is an ETSP (with λ > 0), then H ≠ 0.5.

Let us provide a more specific analysis about the dependence of H on the distribution of the random process. Firstly, to avoid a self-similarity index different from 0.5 due to a mean different from 0, we will work with a symmetric ETSP distribution whose mean is equal to zero and its parameters are a = −0.2, b = 0.2, n = m and λ = δ. Figs 2–5 contain four different situations, where the elevation parameter of the corresponding ETSP distribution varies as follows: λ ∈ {0,0.01,0.05,0.1}. In all the cases, we compare the GHE self-similarity estimation vs. n = m ranging from 10 to 200. Note that for λ = δ = 0.01, the mean self-similarity exponent goes from 0.5 to 0.8; for λ = δ = 0.05, H ranges from 0.55 to 0.75, and finally, for λ = δ = 0.1, the Hurst exponent varies from 0.55 to 0.65.

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Fig 2. Self-similarity index H for symmetric ETSP distributions with parameters a = −0.2, b = 0.2 vs. n = m ranging from 10 to 200.

The figure above corresponds to λ = δ = 0, namely, the TSP case. Segments represent the confidence interval of H at a 90% confidence level, with the point in the segment being the mean value.

https://doi.org/10.1371/journal.pone.0127824.g002

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Fig 3. Self-similarity index H for symmetric ETSP distributions with parameters a = −0.2, b = 0.2 vs. n = m ranging from 10 to 200.

The figure above corresponds to λ = δ = 0.01. Segments represent the confidence interval of H at a 90% confidence level, with the point in the segment being the mean value.

https://doi.org/10.1371/journal.pone.0127824.g003

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Fig 4. Self-similarity index H for symmetric ETSP distributions with parameters a = −0.2, b = 0.2 vs. n = m ranging from 10 to 200.

The figure above corresponds to λ = δ = 0.05. Segments represent the confidence interval of H at a 90% confidence level, with the point in the segment being the mean value.

https://doi.org/10.1371/journal.pone.0127824.g004

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Fig 5. Self-similarity index H for symmetric ETSP distributions with parameters a = −0.2, b = 0.2 vs. n = m ranging from 10 to 200.

The figure above corresponds to λ = δ = 0.1. Segments represent the confidence interval of H at a 90% confidence level, with the point in the segment being the mean value.

https://doi.org/10.1371/journal.pone.0127824.g005

On the other hand, let us also consider a TSP distribution instead of the previous ETSP models for the underlying random process. Thus, if we use a = −0.2, b = 0.2, n = m and λ = δ = 0, then we get that H is approximately constant and very close to 0.5 for all n = m, as we advanced previously (or see both Figs 2 and 3). Similar analysis were also carried out by comparing the self-similarity exponent of ETSP distributions vs. λ (instead of vs. n = m). In this case, we work also with symmetric ETSP distributions (with zero mean) whose parameters are a = −0.2, b = 0.2, n = m ∈ {20,100} and λ = δ ranging from 0.01 to 0.09 by step 0.01. In both cases, the self-similarity exponent of the corresponding time series became more or less stable (close to 0.6 in the case of m = 20, and close to 0.7 in the case of m = 100).

In Fig 6, it is shown the dependence of H on both parameters m = n and λ = δ. Note that in this case, the values of H range from 0.5 to 0.72.

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Fig 6. Level curves for the self-similarity exponent H of an ETSP distribution.

In this case, the parameters were chosen to be a = −0.2, b = 0.2, and different values of m = n and λ = δ.

https://doi.org/10.1371/journal.pone.0127824.g006

Empirical application

In this section, we explore scaling properties for S&P500 stocks as well as for the index itself through analyzing their self-similarity exponents. To do this, let us assume that the distribution of the corresponding series could be properly fitted by means of an ETSP distribution (for daily changes of log returns). Overall, we state that their self-similarity exponents will not provide reliable information about long-term memory properties provided that no additional information about the underlying distribution of such data is previously given. According to that, let us study which stocks from S&P500 index could be properly fitted through an ETSP distribution with explicit parameters given. First of all, let us apply the following notation to refer different self-similarity exponent estimations. Indeed, for a given a stock (or the index itself), let H_Stock denote its self-similarity index, let also H_ETSP denote the self-similarity exponent for an ETSP distribution which properly fits such a stock, and finally, let H_Shuffle denote the self-similarity exponent corresponding to the shuffled time series for that stock. Recall also that the memory component underlying the self-similarity exponent for a given stock could be estimated as H_Stock − H_Shuffle. Further, observe that if a time series is shuffled, then its memory component is removed, so if H_Shuffle becomes different from 0.5, then it yields that this is due to the underlying distribution and not to the memory in such series.

Next, let us verify that the stocks from S&P500 index may be properly fitted through an ETSP distribution with explicit parameters. Note that in such a case, one gets that H_Shuffle = H_ETSP. In this situation, a result such as H_Stock = H_Shuffle = H_ETSP will throw that there is no memory in the series and that the ETSP distribution becomes a good model for stock (log) returns.

Thus, three statistical test will be carried out in the spirit of bootstrapping (see [60]). The statistic is defined as the GHE exponent H. The first of them will consist of checking that H_Shuffle = H_ETSP. Accordingly, the null hypothesis is that $\overline{H_{Shuffle}}=\overline{H_{ETSP}}$, where $\overline{H_{Shuffle}}$ denotes the mean of the self-similarity exponents for 500 shuffled series and $\overline{H_{ETSP}}$ denotes the mean for 500 self-similarity exponents corresponding to ETSP samples generated by means of Monte Carlo simulations. The statistical test is given through a two-tailed p-value, which is defined as On the other hand, the main goal in the second test we perform is to explore persistence in the series, namely, whether H_Stock > H_Shuffle. The null hypothesis in this case is $H_{Stock}=\overline{H_{Shuffle}}$, with $\overline{H_{Shuffle}}$ as previously given, while the alternative hypothesis is $H_{Stock}>\overline{H_{Shuffle}}$. The statistical test is given by a one-tailed p-value, which is given as p₁ = Prob [H_Stock > H_Shuffle].

Moreover, in the third test we want to study anti-persistence in the series, that is, if H_Stock < H_Shuffle. The null hypothesis is $H_{Stock}=\overline{H_{Shuffle}}$, whereas the alternative hypothesis is that $H_{Stock}<\overline{H_{Shuffle}}$, where $\overline{H_{Shuffle}}$ is defined as above. The statistical test is given by a one-tailed p-value, which is defined as p₂ = Prob [H_Stock < H_Shuffle] = 1 − p₁.

Next, let us verify the previous scheme in the actual cases of S&P500 index and its stocks. To do this, we considered data from four years (from March 2011 to April 2014). Firstly, let us test the behavior of S&P500 index. In this way, the optimal fit for the S&P500 index through an ETSP yields the following parameters: a = −0.18, b = 0.17, θ = 0.517, m = 25, n = 25, λ = 0.003 and δ = 0.012. Then the self-similarity index for such an ETSP distribution is calculated by means of Monte Carlo simulations. Thus, we find out that the self-similarity exponent of such an ETSP has a mean equal to H_ETSP = 0.59 (with a standard deviation equal to 0.05), whereas the index itself satisfies that H_Stock = 0.49 and the shuffled index series has a mean of H_Shuffle = 0.54 (with a standard deviation equal to 0.03). The ETSP fit for S&P500 index passed the Kolmogorov-Smirnov goodness-of-fit test. Indeed, the three p-values for S&P500 index were p₀ = 0.12, so H_Shuffle = H_ETSP, p₁ = 0.11 and p₂ = 0.89. Accordingly, we cannot reject any of the null hypothesis of these test provided that a 95% confidence level is considered.

For S&P500 stocks, we find out that only 1.8% of such stocks fail the first test at a confidence level of 95%, so we could conclude that the fitted ETSP distribution does provide a good model for the stocks from the S&P500 index, at least with respect to the self-similarity exponent H (note also that the Kolmogorov-Smirnov goodness-of-fit test was also passed by all the stocks). Hence, the ETSP fit seems to be a good model to test for memory. Next, let us summarize and classify the obtained results for memory purposes in S&P500 stocks throughout the selected time period:

• 0.8% of S&P500 stocks failed the second test (for a confidence level of 95%), so we could conclude that any of such stocks present persistent memory.
• 13.4% of S&P500 stocks failed the third test (also for a confidence level of 95%). An example of this case is Praxair Inc. (PX), for which H_Stock = 0.39, H_Shuffle = 0.51 and H_ETSP = 0.56. This yields empirical evidence regarding memory in the series so that the stocks in this case are anti-persistent.
• 85.8% of the stocks gives similar values for all H_Stock, H_Shuffle and H_ETSP. An example of this situation is O’Reilly Automotive Inc. (ORLY), for which H_Stock = 0.6, H_Shuffle = 0.61 and H_ETSP = 0.61. Accordingly, we could conclude that there is no significant memory in such series, even for self-similarity exponents equal to 0.6.