Permutation Entropy and Information Recovery in Nonlinear Dynamic Economic Time Series

Round 1

Reviewer 1 Report

The paper approaches a particular statistical technique (based on permutation entropy) used to extract information from data associated with nonlinear time series. Most economic processes are too complex to be contained into a unique time invariant econometric model. Therefore, it is necessary to develop tools that allow to econometrically handle complexity. This paper gives a step in such direction, offering a coherent, meaningful and rigorous study that is well complemented with an empirical application. Overall, I perceive this study as a relevant contribution to economic science. 

Author Response

Response: We are happy that this reviewer liked the application of the PE information recovery method that we used.  Although not requested, we have made minor editorial changes to the paper to improve readability.

Reviewer 2 Report

In this paper, the authors propose the use of the permutation entropy metric to assess the evolution of the complexity of stock markets, with a special focus on the Dow Jones Industrial Average. While the paper may be interesting, especially for an econometric audience, I find two major problems.

First of all, the authors state that the Permutation Entropy can be used as a metric of complexity of time series. Please note that this is not completely correct, as the entropy can only assess the degree of “uncertainty” or “predictability”. To be exact, the PE assesses how constraint the dynamics is, taking into account past values. Note that a time series composed of random numbers will have a very high entropy, as it is not predictable; but also a very low complexity.

Authors should refer to several papers, mainly published by O. Rosso, L. Zunino and colleagues, on the used of combined entropy-complexity metrics. For instance:

Zunino, L., Zanin, M., Tabak, B. M., Pérez, D. G., & Rosso, O. A. (2010). Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency. Physica A: Statistical Mechanics and its Applications, 389(9), 1891-1901.

Ribeiro, H. V., Jauregui, M., Zunino, L., & Lenzi, E. K. (2017). Characterizing time series via complexity-entropy curves. Physical Review E, 95(6), 062106.

Secondly, the analysis proposed by the authors is not really new. For instance, they may have a look at the following paper:

Alvarez-Ramirez, J., & Rodríguez, E. (2011). Long-term recurrence patterns in the late 2000 economic crisis: Evidences from entropy analysis of the Dow Jones index. Technological Forecasting and Social Change, 78(8), 1332-1344.

It’s not clear at all what is the novelty of the proposed work.

Author Response

Reviewer number 2

In this paper, the authors propose the use of the permutation entropy metric to assess the evolution of the complexity of stock markets, with a special focus on the Dow Jones Industrial Average. While the paper may be interesting, especially for an econometric audience, I find two major problems.

First of all, the authors state that the Permutation Entropy can be used as a metric of complexity of time series. Please note that this is not completely correct, as the entropy can only assess the degree of “uncertainty” or “predictability”. Taking into account past values, the PE method assesses the level of dynamic constraint, A time series composed of random numbers will have a very high entropy; but a very low complexity.

Authors should refer to several papers, mainly published by O. Rosso, L. Zunino and colleagues, on the used of combined entropy-complexity metrics. For instance:

Zunino, L., Zanin, M., Tabak, B. M., Pérez, D. G., & Rosso, O. A. (2010). Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency. Physica A: Statistical Mechanics and its Applications, 389(9), 1891-1901.

Ribeiro, H. V., Jauregui, M., Zunino, L., & Lenzi, E. K. (2017). Characterizing time series via complexity-entropy curves. Physical Review E, 95(6), 062106.

Response: Thank you for your high quality and helpful comments.

It is certainly possible that a stochastic time series may have high entropy and low complexity. Economic-behavioral processes and systems, the focus of our paper, are typically characterized for being stochastic, dynamic and highly complex. By applying the Bandt and Pompe (2002) PE natural complexity metric on the most widely quoted high-frequency complex economic system (the DJIA) for two analysis periods (1901-2016 and 2000-2016), our high PE results that are close to unity, demonstrate the ability of the PE method to detect the extent of complexity (irregularity) and to discriminate and classify admissible and forbidden states. We have added this to the revised paper. The two proposed papers by Zunino, et al. And Ribiero, et al., are relevant, and have also been added to our paper.

The analysis proposed by the authors is not really new. For instance, they may have a look at the following paper:

Alvarez-Ramirez, J., & Rodríguez, E. (2011). Long-term recurrence patterns in the late 2000 economic crisis: Evidences from entropy analysis of the Dow Jones index. Technological Forecasting and Social Change, 78(8), 1332-1344.

It’s not clear at all what is the novelty of the proposed work.

Response: The suggested paper by Alvarez-Ramirez and Rodriguez (2011) has been added to the revised version of the paper. You are correct that the DJIA index has been already analyzed in that paper using entropy methods. However, we would like to clarify that the entropy approach used by Alvarez-Ramirez and Rodríguez does not involve permutation entropy (PE). To analyze the dynamics of the DJIA, Alvarez-Ramirez and Rodríguez (2011) adopted a non-ordinal probabilistic method known as approximate entropy (ApEn), developed by Pincus in 1991 and rooted in the Eckmann-Ruelle entropy (see Eckmann and Ruelle 1985). PE is a newer-different entropy concept that was developed by Bandt and Pompe (2002), as a natural complexity measure for time series. Unlike ApEn, PE is based on the ordinal structure of the time series and less sensitive to the length of the time series. To the best of our knowledge, our paper is the first one which employs PE involving the DJIA to indicate the information recovery value and applicability of the PE method. We have added this to the revised paper.

REFERENCES

Pincus, S.M. 1991. “Approximate entropy as a measure of system complexity” Proceedings of the National Academy of Sciences of the USA, 88(6): 2297–2301.

Eckmann, J.P. and D. Ruelle. 1985. “Ergodic theory of chaos and strange attractors” Reviews of Modern Physics, 57(3):617–656.

Round 2

Reviewer 2 Report

I'm not very convinced by the answer of the authors, as they essentially added a few references, but not addressed by doubts. In any case, see comments to the Editor.