Analysis of the Romanian Capital Market Using the Fractal Dimension

Round 1

Reviewer 1 Report

 

The authors provide detailed analysis of the Romanian capital market using the fractal dimension. The paper clarify that obtained results are reliable and accurate enough. I feel that it would be intriguing part for the readers.

Overall, the work conducted is worthy of publication in high quality journal. My specific points about this paper can be found below:

Abstract of the paper should be modified by adding new findings.

All the parameters should be added in nomenclature section.

The expression (9) considered should be referenced. Is there any specific reason for assuming such expression?

Improve the quality of the paper by fixing some typos errors.

More discussion on the novelty of the article should be added to conclusions section.

Some recent studies of Fractal and Fractional should be cited at appropriate places.

I believe that the above changes will certainly add value to already well documented contribution by the authors.

 

Author Response

Thanks for the suggestions, I took them into account and made the following changes in the article.

1. Abstract of the paper should be modified by adding new findings.

Abstract: The surrounding reality can be analyzed as a result of the interaction of complex nonlinear dynamic systems. The main objective of the article is to develop and analyse the models that best describe the efficient behaviour of the Romanian capital market that generated the analysed time series. The empirical analysis carried out in this paper do not aim to classify the Romanian market capital as efficient or ineffective, but rather the identification of the degree of deviation from efficiency relative to other markets, respectively the analysis of the dynamics of the degree of deviation over time. To describe the distribution of returns, we focused on the family of generalized hyperbolic distributions, which have statistical properties similar to financial returns. The presence of wide tails in the distributions (of extreme values) suggests the use of statistical tests and measures, for detecting dependencies, which take this behaviour into account. Statistical methods and efficiency indicators are used, such as the Hurst exponent, Taken's theorem and the fractal dimension, which facilitate the detection of the main types of dependencies that could be present in the return’s series, measures that are robust to the heteroscedastic behaviour of the returns. These statistical measures are applied both to the entire period and to sliding windows.

2. All the parameters should be added in nomenclature section.

 

I introduced the abbreviations in the article

 

Abbreviations

d fractional differentiation ordinal

H Hurst coefficient

 is the size of the attractor of interest

, time of sampling

, autocorrelation function

 

 

3. The expression (9) considered should be referenced. Is there any specific reason for assuming such expression?

Regarding the time series typology, we identify the following useful processes to model the financial data generation process:

• stationary processes (zero-order integrated, d=0), characterized by an exponentially decreasing autocorrelation function, corresponding to a short-memory time series. Observations at large intervals of time, one from the other, are independent.
• integrated processes of the first order (d=1), in which case the autocorrelation function decreases linearly, and the observations at large time intervals are not independent.
• fractionally integrated processes [17], characterized by non-zero correlations between observations separated by large time distances, respectively the autocorrelation function decreases slowly towards zero, according to a hyperbolic function. For d ∈ (0, 0.5) the process is stationary and has long persistent memory; autocorrelations are positive and decrease hyperbolically. If d ∈ (−0.5, 0) the process is stationary, being generated series with an anti-persistent behavior. When d ∈ (0.5, 1) the process is non-stationary, being strongly persistent in the long term. Long memory dependencies lead to the invalidation of the random walk model of stock prices and provide arbitrage opportunities on the capital market [18]. Some of the stationary time series show small dependencies between the observations located at large time distances, a fact that denotes a persistent character of the analyzed phenomenon, respectively long memory.

4. Improve the quality of the paper by fixing some typos errors.

I fixed it in the text.

5. More discussion on the novelty of the article should be added to conclusions section.

The work is dedicated to the study of the main statistical properties of stock returns from the perspective of the statistical distribution followed; they usually have a higher distribution and with wider tails than the normal distribution. The presentation of the statistical methodology emphasizes the predictability tests, for the detection of short- and long-term memory in the profitability series. This article highlights an empirical study on the efficiency of the capital market in Romania in the context of 20 developed or developing markets in Europe. The approach to the efficiency of the capital markets is carried out through the prism of moving away from the hypothesis of efficiency and making a ranking of the markets according to the deviations they present. Thus, by using the efficiency context in relative form, the restrictive barrier of the "all or nothing" type, specific to the hypothesis of absolute efficiency, is overcome. The Hurst exponent, Fractal dimension and Taken's theorem are proposed here to be used as measures of efficiency, respectively the efficiency index that considers both short-term and long-term dependencies. Based on the three measures of efficiency, a ranking of the markets is carried out, for a period of 14 years but also on 3 sub-periods delimited according to the economic climate. The obtained results place the Romanian capital market among the markets with the greatest deviations from the weak-form efficiency concept and identify different degrees of efficiency throughout the investigated sub-periods. Making a synthesis in the very vast field of time series analysis and prediction. I particularly insisted on the non-linear analysis of the time series in order to identify the manifestation of chaos according to a methodology that involves going through successive stages. The identification of chaos in other time series corresponding to the manifestations of some processes in the real world and the establishment of possible correlations between the appearance of chaos and qualitative changes in the evolution of the system. The obtained results pave the way for the realization of a prediction algorithm that uses the combined capabilities of neural networks (MLP) and wavelet networks. We empirically demonstrated that, in the case of chaotic time series, the number of inputs of the MLP neural network, for which optimal predictions are obtained, must be approximately equal to the product of the inclusion size and the delay parameter used for reconstruction of the state space, results that will be presented in a subsequent article.

6. Some recent studies of Fractal and Fractional should be cited at appropriate places.

 

I have introduced new references in the article

 

Blackledge, J.M.; Kearney, D.; Lamphiere, M.; Rani, R.; Walsh, P. Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction. MDPI Spec. Issue Math. Econ. Appl. Fract. Calc. 2019, 7, 1–57. Available online: https://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Economics (accessed on 11 August 2022).

West, B.J. Fractional Calculus View of Complexity; CRC Press: Boca Raton, FL, USA, 2020; ISBN 9780367737795

Collatz Conjecture, 2021. Available online: https://en.wikipedia.org/wiki/Collatz_conjecture (accessed on 5 September 2022).

Chaos—Classical and Quantum. Available online: http://www.streamsound.dk/book1/chaos/Chaos/assets/common/ downloads/publication.pdf (accessed on 1 September 2022).

Cootner, P.H. Stock Prices: Random vs. Systematic Changes; EBSCO Publishing: Ipswich, MA, USA, 1962. Available online: http://www.e-m-h.org/Coot62.pdf (accessed on 9 September 2022).

Mifre, J.; Rallis, G. Momentum strategies in commodity futures markets. J. Bank. Financ. 2007, 31, 1863–1886

Zhang, J.; Wu, X.; Yan, R.; Chung, Z. The Liquidity Spillover Effects Between the Stock Index Futures and Spot Under the Fractal Market Hypothesis. Res. Sq. 2021. Available online: https://assets.researchsquare.com/files/rs-933613/v1_covered.pdf?c=1638 462633 (accessed on 9 September 2022)

Lamphiere, M.; Blackledge, J.M.; Kerney, D. Carbon Futures Trading and Short-Term Price Prediction: An Analysis Using the Fractal Market Hypothesis and Evolutionary Computing. Mathematics 2021, 9, 1005.

Denier, J. Millennium Prize: The Navier-Stokes Existence and Uniqueness Problem, 2011. Available online: https: //theconversation.com/millennium-prize-the-navier-stokes-existence-and-uniqueness-problem-4244 (accessed on 7 September 2022).

Blackledge, J. M.; Mosola, N. A Statistically Significant Test to Evaluate the Order or Disorder for a Binary String of a Finite Length. ISSC2020, IEEE UK and Ireland Signal Processing Chapter and IEEE Computational Intelligence Society 2020, Letterkenny Institute of Technology, 11–12 June 2020. Available online: https://arrow.tudublin.ie/engscheleart/311/ (accessed on 9 September 2022).

 

Kind regards,

Authors

 

Reviewer 2 Report

Dear authors,

 In this study the nonlinear analysis of the Romanian capital market and the approach of the chaotic behavior are presented and discussed. The nonlinear analysis of timeseries with various tools is a powerful tool to understand the chaotic dynamics in complex systems.
The classic analysis of the literature is presented but I would like to highlight some weak points:

a)    The methods, the time series of the Romanian capital market and the complexity metrics are presented with a very confused way. Please update the manuscript with a clearer for the readers way (basic theoretical framework, basic methods, data and periods, results, etc)
b)    In Figure 3 the x-axis in Days, 10 days ?, what is the y=axis, price ?
c)    In lines 423-433, where are the data and the periods ?
d)    The timeseries are presented in fig 3, showing strong periods trends. I suggest to authors to exclude the trend by the application of a filter etc. first differences to the signal and then to apply the nonlinear tools. The timeseries should be stationary.
e)    I suggest to be used the mutual information against the autocorrelation function because it looks better the nonlinearity for the choice of delay.
f)    I suggest to calculate the Correlation Dimension and to be used the method of Theiler (J. Theiler, Some comments on the correlations dimensions of 1/f a noise, Phys. Lett. A 155 (1991) 480–493) in order to exclude time correlated states in the correlation integral estimation, thus discriminating between the dynamical character of the correlation integral scaling and the low value saturation of slopes characterizing self-affinity (or crinkliness) of trajectories in a Brownian process.
g)    Please make a graph for table 1 to show the trends of the indices and explain.
h)    Ιt would be good to compare the indices with some major stock exchange for the same period of the Romanian capital market and explain

The authors should present the general theoretical framework and the methodology of the analysis with a more clear, robust and mathematical way.

What is new that this study brings to the international bibliography of nonlinear analysis of complex economical systems?

I advise the authors to reconstruct the study based on the above directions

Please, check the manuscript for errors in the language.  

Finally, I suggest the publication of this study with the major revisions needed.

Best Regards
Reviewer

Author Response

Thanks for the suggestions, I took them into account and made the following changes in the article.

Author Response File: Author Response.pdf

Reviewer 3 Report

First of all, congratulations to the authors for the research work presented in this journal.

This paper describes the analysis of the Romanian capital market using a fractal dimension. The abstract does not clearly indicate what the authors' contributions have been to the study problem addressed in the document. In this way, the advances, results, conclusions, and importance of the research carried out are not appreciated. It is recommended to rewrite the abstract indicating the main contributions of the authors, as well as the conclusions obtained.

The introduction contains the state-of-the-art associated with the study problem. In this case, the main contributions of the authors to the problem analyzed have not been very clear, nor have the advantages and disadvantages compared to other research carried out. As a suggestion, perhaps the elaboration of a table (that shows the contributions of the most important references) can give greater clarity to the state of the art. It is also recommended to increase the number of bibliographical references in the document. This will help to delve into the state of the art.

On the other hand, the document is technically sound, since it contains an analysis of the state of the art related to the problem addressed. It also incorporates different mathematical expressions, diagrams, and graphs associated with the development of the model. It includes some estimated results that support the analysis shown.

Concepts are presented comprehensively. The different figures, tables, diagrams, and schemes facilitate the understanding of the contents presented by the authors in the document. Likewise, the results obtained support the comments made by the authors. All this makes it easy for the reader to follow the paper.

Most of the text incorporated in the conclusions section is more oriented to the reader's discussion than to one's own conclusions. It would be advisable to rewrite this section and, if the authors consider it appropriate, include some comments in the section on the discussion of the results. The number of bibliographical references provided is insufficient. It is advisable to increase their number to improve the state of the art. It is necessary to increase the search for research papers and similar studies.

Author Response

Thanks for the suggestions, I took them into account and made the following changes in the article.

1. The introduction contains the state-of-the-art associated with the study problem. In this case, the main contributions of the authors to the problem analyzed have not been very clear.

The paper is dedicated to the study of the main statistical properties of stock returns from the perspective of the tracked statistical distribution; they usually have a larger distribution and with wider tails than the normal distribution. The presentation of the statistical methodology emphasizes the predictability tests, for the detection of short- and long-term memory in the profitability series. The approach to the efficiency of the capital markets is carried out through the lens of moving away from the hypothesis of efficiency and creating a hierarchy of the markets according to the deviations they present. Thus, by using the efficiency context in relative form, the restrictive barrier of the "all or nothing" type, specific to the absolute efficiency hypothesis, is overcome. The Hurst exponent, fractal dimension and Taken's theorem are proposed here to be used as efficiency measures, respectively the efficiency index that considers both short-term and long-term dependencies.

 

2. Recommended to increase the number of bibliographical references in the document.

I have introduced new references in the article

Blackledge, J.M.; Kearney, D.; Lamphiere, M.; Rani, R.; Walsh, P. Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction. MDPI Spec. Issue Math. Econ. Appl. Fract. Calc. 2019, 7, 1–57. Available online: https://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Economics (accessed on 11 August 2022).

West, B.J. Fractional Calculus View of Complexity; CRC Press: Boca Raton, FL, USA, 2020; ISBN 9780367737795

Collatz Conjecture, 2021. Available online: https://en.wikipedia.org/wiki/Collatz_conjecture (accessed on 5 September 2022).

Chaos—Classical and Quantum. Available online: http://www.streamsound.dk/book1/chaos/Chaos/assets/common/ downloads/publication.pdf (accessed on 1 September 2022).

Cootner, P.H. Stock Prices: Random vs. Systematic Changes; EBSCO Publishing: Ipswich, MA, USA, 1962. Available online: http://www.e-m-h.org/Coot62.pdf (accessed on 9 September 2022).

Mifre, J.; Rallis, G. Momentum strategies in commodity futures markets. J. Bank. Financ. 2007, 31, 1863–1886

Zhang, J.; Wu, X.; Yan, R.; Chung, Z. The Liquidity Spillover Effects Between the Stock Index Futures and Spot Under the Fractal Market Hypothesis. Res. Sq. 2021. Available online: https://assets.researchsquare.com/files/rs-933613/v1_covered.pdf?c=1638 462633 (accessed on 9 September 2022)

Lamphiere, M.; Blackledge, J.M.; Kerney, D. Carbon Futures Trading and Short-Term Price Prediction: An Analysis Using the Fractal Market Hypothesis and Evolutionary Computing. Mathematics 2021, 9, 1005.

Denier, J. Millennium Prize: The Navier-Stokes Existence and Uniqueness Problem, 2011. Available online: https: //theconversation.com/millennium-prize-the-navier-stokes-existence-and-uniqueness-problem-4244 (accessed on 7 September 2022).

Blackledge, J. M.; Mosola, N. A Statistically Significant Test to Evaluate the Order or Disorder for a Binary String of a Finite Length. ISSC2020, IEEE UK and Ireland Signal Processing Chapter and IEEE Computational Intelligence Society 2020, Letterkenny Institute of Technology, 11–12 June 2020. Available online: https://arrow.tudublin.ie/engscheleart/311/ (accessed on 9 September 2022).

 

3. The main contributions of the authors to the analysed problem.

Based on these references, the contribution of the authors will be highlighted.

The fractal markets hypothesis emphasizes the impact of market liquidity and the investment period on market equilibrium. Unlike the theory of efficient markets, where the focus is on market efficiency, in the theory of fractal markets, the focus is on market stability. In such a market, investors assume the same level of risk, which leads to similarities in the distribution of returns regardless of the time horizon allocated to the investment. Due to these similarities, the association between the capital market and the notion of fractal is made. Economic analysts admit that efficiency in absolute form remains an ideal that is difficult to achieve in practice, and empirical studies can only determine how closely a market approaches this ideal. Starting from the empirical evidence that supports this statement, a step forward in the theory of efficient markets is the evolutionary approach to informational efficiency, by investigating the degree of efficiency of markets from the perspective of dynamics over time. The main empirical contribution of this article consists in the analysis of the degree of relative efficiency of 5 Romanian capital markets, both from the perspective of time dynamics and by using several measures to quantify the deviations from the efficiency hypothesis. Some of the measures used to quantify the degree of efficiency were calculated for the entire period, but also on sliding windows, to ensure greater robustness of the estimated value. Among the measures used is a generalized measure, proposed relatively recently in the literature [37-46], which incorporates both the deviations that appear as a result of the existence, in the return series, of some short-term and some long-term dependencies. The rankings were made based on three measures, namely the Hurst exponent, the fractal dimension and the Takens generalized measure. The Hurst exponent identifies the presence of long memory, the fractal dimension is an indicator of local memory, and the generalized measure includes, in addition to the two indicators, the first-order autocorrelation coefficient (measures short-term memory). We analyze the predictability of stock market returns based on deviations from the walking pattern random, generated by the existence of short-term, linear or non-linear dependencies, respectively long-term dependencies. Mainly, long memory can be described, based on the self-similarity property, by the fractal dimension; this represents a local characteristic of the time series. Multifractality has become a suitable scientific framework for the study of efficiency. From a methodological point of view, in order to identify whether the considered Romanian markets present deviations from the weak form of efficiency, the Hurst exponent, the fractal dimension and the generalized measure of the degree of efficiency are used as measures of efficiency. Long-term dependencies are estimated based on the Hurst exponent, calculated both on the whole period and on overlapping windows. The overlapping window estimation methodology involves the use, at each step, of a fixed number of observations, namely 300, and the estimation of the Hurst exponent for that window. At each step, a new window is formed by serially introducing the next return and discarding the oldest observation. In this way, several values ​​of the Hurst exponent are obtained, namely the total number of returns minus the length of the window. The Hurst exponent used in the comparisons is given by the average between the exponent estimated over the entire period and the median value of the set of estimates obtained from the overlapping windows procedure. The fractal dimension is calculated as the average of the estimated value. The generalized measure, introduced in the article, is based on long-term, short-term dependency sizes and fractality sizes, respectively, the Hurst exponent, the first-order autocorrelation coefficient, and the fractal dimension. In the calculation of the generalized measure, the following are considered: the average of the Hurst exponent estimators over the whole period and on overlapping windows, the average of the fractal dimension estimators and the first-order autocorrelation coefficient. In making the index rankings according to the degree of deviation from the concept of efficiency, the deviations in absolute value of the Hurst exponent, respectively of the fractal dimension, from the corresponding value of an efficient market (0.5 for the Hurst exponent, and 1.5 for the fractal dimension) were used, respectively the generalized measure. The obtained results pave the way for the realization of a prediction algorithm that uses the combined capabilities of neural networks (MLP) and wavelet networks. We empirically demonstrated that, in the case of chaotic time series, the number of inputs of the MLP neural network, for which optimal predictions are obtained, must be approximately equal to the product of the inclusion size and the delay parameter used for reconstruction of the state space, results that will be presented in a subsequent article.

 

Kind regards,

Authors

Round 2

Reviewer 2 Report

Thank you authors for the revized manuscript.

I satisfy with the updated manuscript.

I accept the manuscript for publication in the updated form.

My best wishes

Reviewer

 

Author Response

Thanks for the suggestions, I took them into account and made the following changes in the article.

 

   Kind regards,

   Authors

Reviewer 3 Report

The authors have incorporated in the document the different observations and comments made by the reviewers. In this way, the state-of-the-art has been improved and new bibliographic references have been incorporated that provide greater depth to the study. Likewise, details have been included throughout the text that provides different clarifications to the reviewers. However, it is advisable to present a table that collects the main contributions of the most significant bibliographical references to the subject of study. All this will allow us to improve the state-of-the-art.

As for the figures, it would be advisable to homogenize the size of the data and numbers on the diagrams.

Author Response

Thanks for the suggestions, I took them into account and made the following changes in the article.

 

   Kind regards,

   Authors

Author Response File: Author Response.pdf