{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:58Z","timestamp":1760243278127,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2014,5,19]],"date-time":"2014-05-19T00:00:00Z","timestamp":1400457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 H\u00a8older 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 H\u00a8older exponent. These procedures enhance the capacity to identify the occurrence of financial crashes.<\/jats:p>","DOI":"10.3390\/e16052768","type":"journal-article","created":{"date-parts":[[2014,5,19]],"date-time":"2014-05-19T11:13:29Z","timestamp":1400498009000},"page":"2768-2788","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Market Efficiency, Roughness and Long Memory in PSI20 Index Returns: Wavelet and Entropy Analysis"],"prefix":"10.3390","volume":"16","author":[{"given":"Rui","family":"Pascoal","sequence":"first","affiliation":[{"name":"School of Economics, University of Coimbra, Avenida Doutor Dias da Silva 165, 3004-512 Coimbra, Coimbra, Portugal"}]},{"given":"Ana","family":"Monteiro","sequence":"additional","affiliation":[{"name":"School of Economics, University of Coimbra, Avenida Doutor Dias da Silva 165, 3004-512 Coimbra, Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2014,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.physa.2012.08.003","article-title":"Measuring capital market efficiency: Global and local correlations structure","volume":"392","author":"Kristoufek","year":"2013","journal-title":"Physica A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.eneco.2013.12.001","article-title":"Commodity futures and market efficiency","volume":"42","author":"Kristoufek","year":"2014","journal-title":"Energ. Econ"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1086\/294743","article-title":"The Behavior of Stock-Market Prices","volume":"38","author":"Fama","year":"1965","journal-title":"J. Bus"},{"key":"ref_4","first-page":"41","article-title":"Proof That Properly Anticipated Prices Fluctuate Randomly","volume":"6","author":"Samuelson","year":"1965","journal-title":"Ind. Manag. Rev"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Brockwell, P., and Davis, R. (2002). Introduction to Time Series and Forecasting, Springer. [2nd ed].","DOI":"10.1007\/b97391"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physa.2004.03.072","article-title":"Dynamical scenario for nonextensive statistical mechanics","volume":"340","author":"Tsallis","year":"2004","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3069","DOI":"10.1016\/j.physa.2012.01.033","article-title":"Tsallis statistics and magnetospheric self-organization","volume":"391","author":"Pavlos","year":"2012","journal-title":"Physica A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1829","DOI":"10.1016\/j.physa.2009.12.020","article-title":"Tsallis\u2019 q-triplet and the ozone layer","volume":"389","author":"Ferri","year":"2010","journal-title":"Physica A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"39001","DOI":"10.1209\/0295-5075\/102\/39001","article-title":"Non-extensive triplet in geological faults system","volume":"102","author":"Scheerer","year":"2013","journal-title":"Europhys. Lett"},{"key":"ref_10","first-page":"231","article-title":"A fast MATLAB program to estimate the multifractal spectrum of multidimensional data: Application to fractures","volume":"37","author":"Rostirolla","year":"2011","journal-title":"Comput. Geosci"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1140\/epjb\/e2006-00205-y","article-title":"A nonextensive approach to the dynamics of financial observables","volume":"55","author":"Moyano","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_12","first-page":"91","article-title":"Wavelet Techniques in Multifractal Analysis","volume":"72","author":"Jaffard","year":"2004","journal-title":"Fractal Geometry and Applications: A Jubilee of Beno\u00eet Mandelbrot"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.1109\/TSP.2007.896269","article-title":"Multifractality tests using bootstrapped wavelet leaders","volume":"55","author":"Wendt","year":"2007","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1250048","DOI":"10.1142\/S0219691312500488","article-title":"An entropy based in wavelet leaders to quantify the local regularity of a signal and its application to analize the Dow Jones index","volume":"10","author":"Rosenblatt","year":"2012","journal-title":"Int. J. Wavelets Multiresolut. Inf. Process"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1029\/WR014i003p00491","article-title":"Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst phenomenon","volume":"14","author":"McLeod","year":"1978","journal-title":"Water Resour. Res"},{"key":"ref_16","first-page":"153","article-title":"Defining and Measuring Long-Range Dependence","volume":"11","author":"Cutler","year":"1997","journal-title":"Nonlinear Dynamics and Time Series: Building a Bridge between the Natural and Statistical Sciences"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.2307\/2938368","article-title":"Long-Term Memory in Stock Market Prices","volume":"59","author":"Lo","year":"1991","journal-title":"Econometrica"},{"key":"ref_18","first-page":"140","article-title":"Contribution to discussion of paper by A.J. Lawrance and N.T. Kottegoda","volume":"34","author":"Cox","year":"1977","journal-title":"J. R. Stat. Soc"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1111\/j.1467-9892.1983.tb00371.x","article-title":"The estimation and application of long memory time series model","volume":"4","author":"Geweke","year":"1983","journal-title":"J. Time Anal"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1061\/TACEAT.0006518","article-title":"Long term storage capacity of reservoirs","volume":"116","author":"Hurst","year":"1951","journal-title":"Trans. Am. Soc. Civ. Eng"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"703","DOI":"10.2307\/1913610","article-title":"A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix","volume":"55","author":"Newey","year":"1987","journal-title":"Econometrica"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1214\/aos\/1176324317","article-title":"Gaussian Semiparametric Estimation of Long Range Dependence","volume":"23","author":"Robinson","year":"1995","journal-title":"Ann. Stat"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1364\/JOSAA.7.001055","article-title":"Estimating fractal dimension","volume":"7","author":"Theiler","year":"1990","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1093\/biomet\/80.1.246","article-title":"On the performance of box-counting estimators of fractal dimension","volume":"80","author":"Hall","year":"1993","journal-title":"Biometrica"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1023\/A:1021728614555","article-title":"Highly Robust Variogram Estimation","volume":"30","author":"Genton","year":"1998","journal-title":"Math. Geol"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/BF01619355","article-title":"A regularity statistic for medical data analysis","volume":"7","author":"Pincus","year":"1991","journal-title":"J. Clin. Monitor"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"13709","DOI":"10.1073\/pnas.0405168101","article-title":"Irregularity, volatility, risk, and financial market time series","volume":"101","author":"Pincus","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1103\/PhysRevLett.80.53","article-title":"Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems","volume":"80","author":"Lyra","year":"2004","journal-title":"Phys. Rev. Lett"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2497","DOI":"10.3390\/e12122497","article-title":"the Incomplete Information Theory","volume":"12","author":"Darooneh","year":"2010","journal-title":"Entropy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.physa.2006.10.099","article-title":"Non-extensive behavior of a stock market index at microscopic time scales","volume":"377","author":"Cortines","year":"2007","journal-title":"Physica A"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Jensen, A., and la Cour-Harbo, A. (2001). Ripples in Mathematics, Springer.","DOI":"10.1007\/978-3-642-56702-5"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Percival, D., and Walden, A. (2000). Wavelet Methods for Time Series Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511841040"},{"key":"ref_34","unstructured":"Mallat, S. (2009). A Wavelet Tour of Signal Processing, the Sparse Way, Elsevier."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Genc\u00b8ay, R., Sel\u00e7uk, F., and Whicther, B. (2002). An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press.","DOI":"10.1016\/B978-012279670-8.50004-5"},{"key":"ref_36","unstructured":"Daubechies, I. (1998). Ten Lectures on Wavelets, SIAM. [1st ed]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1214\/11-STS370","article-title":"Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data","volume":"27","author":"Gneiting","year":"2012","journal-title":"Stat. Sci"},{"key":"ref_38","unstructured":"Zivot, E., and Wang, J. (2006). Modeling Financial Time Series with S-PLUS, Springer."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/713665670","article-title":"Empirical properties of asset returns: Stylized facts and statistical issues","volume":"1","author":"Cont","year":"2001","journal-title":"Quant. Financ"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0304-4076(95)01732-1","article-title":"Long Memory Processes and Fractional Integration in Econometrics","volume":"73","author":"Baillie","year":"1996","journal-title":"J. Econom"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0304-4076(95)01749-6","article-title":"Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity","volume":"74","author":"Baillie","year":"1996","journal-title":"J. Econom"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/5\/2768\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:11:33Z","timestamp":1760217093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/5\/2768"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,5,19]]},"references-count":41,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2014,5]]}},"alternative-id":["e16052768"],"URL":"https:\/\/doi.org\/10.3390\/e16052768","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,5,19]]}}}