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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this study, features of the financial returns of the PSI20index, related to market efficiency, are captured using wavelet- and entropy-based techniques. This characterization includes the following points. First, the detection of long memory, associated with low frequencies, and a global measure of the time series: the Hurst exponent estimated by several methods, including wavelets. Second, the degree of roughness, or regularity variation, associated with the Hölder exponent, fractal dimension and estimation based on the multifractal spectrum. Finally, the degree of the unpredictability of the series, estimated by approximate entropy. These aspects may also be studied through the concepts of non-extensive entropy and distribution using, for instance, the Tsallis q-triplet. They allow one to study the existence of efficiency in the financial market. On the other hand, the study of local roughness is performed by considering wavelet leader-based entropy. In fact, the wavelet coefficients are computed from a multiresolution analysis, and the wavelet leaders are defined by the local suprema of these coefficients, near the point that we are considering. The resulting entropy is more accurate in that detection than the Hölder exponent. These procedures enhance the capacity to identify the occurrence of financial crashes.

The purpose of this paper is to analyze two main issues concerning the Portuguese Index PSI20 daily returns, in the period from 2000 to 2013. The first issue is market efficiency (see Kristoufek and Vorvsda [

Long memory reflects a tendency for a slow decay of the magnitude of the time series correlations as a function of lag size, but still preserving stationarity. In a strict sense, it is defined as reflecting a trend-like behavior, that is, persistence (positive long memory). In a broader sense, it may reflect either persistence or anti-persistence, that is, a switching behavior more pronounced than that of a random process (negative long memory); see [

While long memory, measured by the Hurst coefficient, H, is a global characteristic of the series, roughness, measured by the fractal dimension, D, is a local one. For a self-affine process, given by:

it is verified that D + H = 2 (see, for instance, [

A generalization is given by multifractal processes, given by:

where

so that a linear scaling is attained.

The definition of entropy as a measure of uncertainty, or lack of information, is used not only to measure unpredictability, but also to reflect indirectly the degree of roughness in the path of the series.

As an alternative in the analysis of market efficiency, we will use the concept of the q-triplet, created by Tsallis [

The second issue is local regularity, which is related to the identification of crisis events. Financial time series evolve showing patterns, such as time varying volatility and abrupt changes. Irregularity presented by a signal gives information about its behavior. The characterization of irregularity is given by the quantification of the local regularity of a function,

The rest of this paper is organized as follows. In Section 2, we present the methods for studying market efficiency, using the concepts of long memory, unpredictability and path roughness (Section 2.1) and the concept of q-triplet (Section 2.2). The treatment of the wavelet approach for measuring irregularity is given in Section 3. Results are presented in Section 4. We provide a brief conclusion in Section 5.

Analysis of the existence of efficiency in the financial markets is an important issue in financial analysis. A capital market is considered efficient if prices, at each moment, reflect all relevant information. Three types of efficiency are considered according to the degree of information incorporated: weak efficiency if information includes only past prices; semi-strong efficiency if information includes also the information publicly available in the market; and strong efficiency if it includes all information, public or private.

We analyze three aspects of the return series, which result from the existence of efficiency. The empirical testing of those aspects correspond to the check for the existence of efficiency.

The assurance of efficiency in the market is attained by an automatic elimination of arbitrage opportunities. The absence of arbitrage implies that there will be no long memory, described as a power decay of correlations, in the return series—the first aspect that characterizes the return series. As a consequence of the absence of arbitrage opportunities, it is expected that the price series are modeled as a martingale and the corresponding returns as a martingale difference that is unpredictable—the second aspect of the return series. This implies an erratic behavior of prices, which is quantified as a high degree of roughness—the third aspect of the return series.

A long memory process was originally defined and motivated, respectively, by McLeod and Hipel [

where

In what follows, we assume that the parameter,

An alternative definition of a long memory process is attained by the asymptotic characterization of autocorrelations (see [

where _{1} is a slow variation function (that is, a measurable function, which is positive in a neighborhood of ∞ and for which:

A third alternative definition may be stated in the frequency domain; see [

where _{2} is a slow variation function, λ denotes a frequency and

_{1}

_{t}

_{3}

In alternative definitions of long memory, we consider asymptotic relations based on power functions, either in the time or in the frequency domain. By applying a logarithm transformation on the variables, we obtain linear regressions, which may be fitted by a least squares approach. This is the mechanism taken to build several estimation methods for the Hurst coefficient,

Considering a long memory process in the form
_{t}_{t}

where _{ε}(λ) and _{f}

The periodogram estimator is obtained after (

The

where
_{T}

where {_{t}_{T}_{T}

Other approaches may be taken, for instance, the Whittle estimator [

The fractal dimension is a measure of roughness, and in opposition to the Long memory characterization, it measures the local memory of the series (Kristoufek and Vorvsda [

(see Theiler [

so that the dimension is given by:

This is a local quantity from which a global definition of the fractal dimension can be found by averaging.

(1) The classical box-counting dimension is defined as follows. We take a partition of the state space in a grid where each box has size

(2) The Hall–Wood estimator is a version of the box-counting estimator, this being obtained from (^{2}. Therefore, the dimension is given by:

The Hall–Wood estimator is based on an ordinary least squares regression fit of

where ^{k}_{2}(n).

and

(see Hall

(3) The Genton estimator is based on the variogram given by 2γ_{2} (

for which we have γ_{2}(^{α}

The ordinary least squares regression fit of:

on

Remark that _{ℓ}

The mean squared error of the estimator is minimized for

When the method of moments estimator

which is approximately equal to
_{h}_{h}_{i}_{j}_{i},x_{j}_{i} – x_{j} = h

As Kristoufek and Vosvrda [

is built by making

where:

being:

and

As referred to by Pincus

Typically, it is chosen ^{m}^{m}

The concept of the q-triplet created by Tsallis [

This indicator allows one to stress the power-law sensitivity to initial conditions (Lyra and Tsallis [

In the exponential deviation case, we have, ^{λ1}^{t}_{1} is the Liapunov exponent. The power-law sensitivity to initial conditions is given by:

It is a generalization of the classical exponential case; the limit case as

The entropic index, _{i}

so that

Considering
^{ }^{f }^{(}^{α}^{)}. Note that

and

We take the _{+∞} (resp. _{−∞}). Note that for a given
^{−1}. Then, recalling that

and consequently:

Therefore,
_{+∞}, and it can become at most a splitting of order ℓ_{−∞}, we can express (

On the other hand, after (

Finally,
_{sens}

This indicator is defined in the context of the relaxation of an observable variable, _{t}

which behaves as a function of time

where

where

In a time series context, the relaxation variable is the autocorrelation function of

To estimate _{rel}_{q} C_{q}_{rel}

This is the

(the normalizing condition, so that we have a probability distribution) and:

(the mean value under the escort distribution

In the financial context (see Queirós _{q}

In this case, we attain the q-Gaussian distribution, with density function:

where:

and

_{i}_{j}

_{q} as q

_{t} is a Wiener process and p

By reparametrization, it can be seen that the q-Gaussian distribution is, in fact, a t-student distribution where,

Parameters can be estimated by maximum likelihood or minimizing the mean square deviation between this distribution and the empirical distribution (see Cortines

Alternatively, the following procedure may be applied. The range of values for _{i}_{i}

Therefore, we have or

where ln_{q}_{q}_{q}

The q-Gaussian distribution may be seen as the one associated with a stationary state. The q-statcoefficient reflects the sensibility of the volatility to the occurrence of higher variations, such as crashes, resulting from self-organization in the market (see Pavlos

A wavelet transform is a possible representation for a time series, so that the information given by the data can be captured in a clarified way. In order to capture the features of the time series, a basic function, called the mother wavelet, is used. The mother wavelet is shifted and stretched, so that the different frequencies, at different times, can be revealed and the events that are local in time are captured. This enables the wavelet transform to study nonstationary time series.

Wavelets can be used to decompose a time series showing its different components. The analysis using wavelets converts the original signal into different domains with different levels of resolution, so that the time series can be analyzed and processed. In fact, while Fourier transforms decompose the signal as a linear combination of sine and cosine functions, the wavelet transform explains the signal as a sum of flexible functions that allow a localization in frequency and time. Depending on the purposes of the study, we have different wavelet transforms: continuous and discrete. In particular, the discrete wavelet transform (DWT) allows one to decompose a time series, originating a set of coefficients that are obtained using the shifted and stretched versions of the mother wavelet. The DWT of a time series can be a way to represent a signal using a small number of terms. General references on wavelet transforms include, among others, [

The multiresolution pyramidal decomposition allows one to decompose a signal into detailed and approximated signals. The detailed signals express the high frequency components, while the approximated signals express the low frequency components. In order to check for the regularity, we should consider an orthogonal decimated discrete wavelet transform with fast decay derivatives and an appropriate number of vanishing moments.

In order to quantify the local regularity of a function,

Jaffard [

Consider Ψ_{0}, a real valued function with compact support and:

Define the number

_{0}.

Let us consider translations and dilations of Ψ_{0} :

The set {Ψ_{j,k}^{2}(). Given a signal, _{X}

and the signal can be written as:

Assuming that Ψ_{0}(_{j,k}^{j}^{j}

The wavelet leaders are:

Then, for each level, _{0} ∈ _{0}, _{j,k }_{j}_{0}). Let 3_{j}_{0}) be the interval [(^{j}^{j}_{0} is:

To study the local regularity of a time series, Rosenblatt _{1}, y_{2},…, _{m}_{1},…, _{m}_{i}_{i}_{2}(_{i}

Given a bounded function, _{0} ∈ _{f}_{x}_{0}, in order to present the pointwise leader entropy Considering

where _{i}_{0}) is the wavelet leader coefficient for _{0} and resolution level _{x}_{0} = (_{1},…, _{m}_{0} ∈ _{f}

(if _{i}_{2} (_{i}_{0}, belong to the highest resolution level (indicating more roughness), then the wavelet coefficients for _{0} are equal and _{f}_{0}) is maximum (equal to log_{2}(m)). If, on the other hand, the wavelet coefficients for the neighborhood of _{0} are near zero, then _{f}_{0}) ≈ 0.

In this section, we report some numerical experiments, related to the market efficiency topics and local regularity presented in the paper, applied to the Portuguese PSI20 Index data.

This is a stock market index for the twenty most relevant assets, traded on the Euronext Lisbon, a small dimension market. These assets are selected and weighted according to their market capitalization and liquidity. The composition of the index is revised regularly. From the twenty stocks, a small number hold the majority of market capitalization. PSI20 has a non-negligible correlation with the main European stock markets.

The data was collected from the Yahoo Finance publicly available database. We store settlement prices from 2000 to 2013. Let

Fractal dimension results are presented in

We consider different approaches, such as Hall–Wood, Genton and box-counting estimators (see [

The reference value for the fractal dimension is 1.5, which stands for the absence of either local persistence or local anti-persistence. If the fractal dimension is greater (resp. smaller) than 1.5, it means that there is local anti-persistence (resp. persistence), and the series path is rougher (resp. less rough) than in the reference case.

For log-prices, the values we found are slightly less than 1.5, indicating that there is no significant local persistence. As for the estimated series of integrated variance, we obtain values close to unity, which means that there is strong local persistence.

The Hurst coefficient was computed with different approaches, such as Geweke Porter-Hudak, periodogram and R/Sestimators; see [

The reference value for the Hurst coefficient is 0.5, which stands for the absence of either positive long memory or negative long memory. If the Hurst coefficient is greater (resp. smaller) than 0.5, then there is positive (resp. negative) long memory in the series.

In our data, there is long memory in the squared return series (representing volatility), but not in the returns series (as observed in general, in the empirical literature; see [

In

For the q-triplet, the reference value is one (see, for instance, [

In our data, the q-triplet assumes the following results: _{rel}_{rel}_{stat}_{sens}_{sens}_{sens}

There is an inverse relation between the pointwise Hölder exponent and the pointwise wavelet leader entropy, as we pointed out before. We computed the wavelet coefficients considering an orthogonal decimated discrete wavelet transform; our mother wavelet is a Daubechies, with three vanishing moments. After computing the wavelet leader coefficients for the return series, we estimate the temporal evolution of the pointwise wavelet leader entropy (see

The entropy is considered to be the maximum when we have the most uncertain situation. Wavelet leader values near _{2} (8) (eight is the number of scales from the wavelet decomposition) indicate high regularity in the signal, while values near zero indicate low regularity. The temporal evolution of regularity allows one to identify a crisis; the financial crisis of 2008 and subsequent local minimums in PSI20 returns. Those dates are indicated in _{2} 8 for the pointwise wavelet leader entropy, we expect an irregularity in the signal).

In our estimations, we find that the PSI20 returns series is highly unpredictable, rougher than a normally distributed series and has no long memory (but has persistent volatility). These characteristics are typical findings in an efficient market. The analysis of local regularity, using wavelet leaders, allows one to identify the moments of a crash in the Portuguese market as those where a peak, in the degree of irregularity, is attained. Therefore, this technique may be seen as a means to identify those kinds of events.

The authors would like to thank the two anonymous reviewers for their helpful comments.

Both authors contributed to the initial motivation of the problem, to research and estimation process, and to the writing. Both authors read and approved the final manuscript.

The authors declare no conflicts of interest.

PSI20index returns.

Approximate entropy comparison.

The multifractal spectrum.

The wavelet leader entropy.

The fractal dimension.

PSI20 log-prices | Estimated Integrated variance of PSI20 log-prices | |
---|---|---|

Hall–Wood | 1.43 | 1.00 |

Genton | 1.43 | 1.03 |

Box-count | 1.38 | 1.04 |

The Hurst coefficient.

Hurst | PSI20 returns | Squared PSI20 returns |
---|---|---|

GPH | 0.5483 (conf_{lo}_{hi} |
0.799 |

Periodogram | 0.528 | 0.7456 |

0.5774 | 0.8845 |