Maybe I'm missing a broad swath of empirical literature, but I'd be curious why this doesn't have more than cites. I'd love to hear from someone what I am missing that makes this work much less important than I think it is. I am sorry I'm just beside myself right now. I think Buchen and Kelly, whoever they are, got cheated out of a really good paper but maybe I'm saying that because I got attached to an idea I came up with myself. Please log in or register to add a comment.
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I don't know about the reasons why this particular approach was not followed. I do know, however, quite about about the criticisms one gets in the rational inattention literature, a part of which is also based on Shannon entropy. One concern people have is that they do not understand the basis for using that particular measure of entropy.
It's derived from information theoretic considerations, so why should it be relevant to particular economic settings, where considerations about channel capacity and optimal information transmission play no role. Why should we use Shannon entropy, rather than any of the half-dozen other entropy measures? At the same time, everyone is intimately familiar with central limit theorems. And while they are, of course, aware that the following is false, they nonetheless often seem to make intuitive assessments as if the central limit theorems implied that all probability distributions are normal.
Another reason might be the fact that the authors are not very well known. If you decide to build on somebody else's research, you are taking a risk. The higher the chance the research you're building on will be accepted, the higher the chance your own work will be accepted. And that chance is higher for better-known authors. I appreciate the response and I think it is a good one and so I upvoted. But why choose the more restrictive constraint. The maximum entropy distribution with a set mean and set standard deviation is a Gaussian, so if the constraints can imply that, you will get a Gaussian out of the model.
Meanwhile Shannon entropy is the natural entropy to use because AFAIK, it is the only entropy that basically doesn't when applied to a distribution doesn't allow any other "spurious" constraints to be embedded in it. Given a set of constraints, it satisfies them and is as natural as possible to any other constraint that could be applied but is not applied. But even if you don't like Shannon entropy, use another entropy if you can get an equation out of it.
Either way, there is no reason to restrict oneself to a Gaussian.
We then compare the KS statistic observed for the empirical data when compared to a fitted power law distribution with the KS statistic obtained for the synthetic data when compared to a fitted power law distribution. To determine whether the distribution is consistent with exponential decay, we perform a parallel analysis fitting the data to an exponential distribution instead of a power law probability distribution. It contains the following data:. We depict in blue the cumulative distribution function of the positive component of the return distributions for American Express for a time lag of seconds. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. View Wishlist. Power laws in economics and finance.
I do agree with the fact that the authors are not well known probably is a big reason. However, there have been a decent number of theory papers building on this but I could hardly find one empirical paper.
This is not my area, and I suspect that someone who does work in this area can address maximum entropy more directly. But, at a very general level, the success of a model or a method depends on the insights that it can express. For most purposes, brownian motions or brownian motions with jumps have a lot of convenient modeling properties that are useful for modeling purposes even though we don't believe they are the most accurate model for returns or prices. Heck, many of the most successful asset pricing papers assume time is discrete, utility is exponential, etc. Our findings may inform the development of models of market behavior across varying time scales.
Complex movements in stock market prices affect the personal fortunes of people around the globe [ 1 — 5 ]. An ability to more accurately quantify and predict such changes would allow us to gain more insights into how financial crises arise [ 6 ] and provide greater empirical basis for the development of theories of financial market behavior [ 7 — 13 ].
The financial markets were however one of the earliest sources of large scale datesets on human behaviour, where such data have recently become the focus of the new field of computational social science [ 14 — 24 ]. A vast amount of data on financial decisions made in stock markets is therefore available [ 25 — 29 ]. Previous studies have shown that distributions of returns observed in empirical data are consistent with power law decay [ 30 — 42 ], in contrast with widely used models that assume Gaussian behavior of these returns.
Power law behavior has also been observed in other economical and financial sectors of society [ 43 , 44 ]. Changes in stock market prices can occur at a range of different time scales. Here, we analyze a large dataset of stocks forming the Dow Jones Industrial Average DJIA at a second-by-second resolution for a range of different time scales in order to quantify the distribution of returns.
We provide evidence that while the distribution of returns exhibits power law behavior at small time scale, exponential behavior is observed at larger time scales. We find analogous results when restricting our analysis to volatile trading periods. Our findings could help to gain insight into changes in stock market prices in shorter periods and longer periods and provide further empirical basis for the development of new models of market behavior. Our dataset covers the period from 2 January to 30 July comprising a total of trading days.
Fig 1 shows the various components of the DJIA. As five stocks were replaced during this period, we focus on the 25 components that were consistently part of the DJIA between 02 January and 30 July Dashed vertical lines correspond to changes in the stocks forming the DJIA. In our analysis, we focus on the 25 stocks that were part of the DJIA during the period of analysis. Stocks are labelled using ticker symbols that uniquely identify the company name, as used by the stock exchange.
We define returns as the relative logarithmic change in price of a given stock i at a given time t :. We compute the standardized distribution of the returns for the 25 components of the DJIA that we consider. We conduct separate analyses of the cumulative distribution function CDF of the positive and negative component of the distribution of returns.
Note that the empirical distribution strongly deviates from the Gaussian distribution and provides initial evidence for power law behavior. We perform a statistical analysis to check the consistency of the tails of the empirical distributions with power law behavior across different time scales, as proposed by Clauset, Shalizi and Newman [ 45 ] and detailed in the Methods section. We build returns distributions for the 25 stocks of the DJIA for different time lags across the full period of analysis.
We standardize each distribution by subtracting the mean return from each observation and dividing by the standard deviation. We depict in blue the cumulative distribution function of the positive component of the return distributions for American Express for a time lag of seconds. We depict in red the positive tail of a Gaussian distribution with mean zero and standard deviation one. We observe a strong deviation of the empirical distribution from the Gaussian distribution.
Instead, visual inspection of the distribution tail reveals consistency with a linear relationship on a log-log scale. This provides initial evidence for possible power law behavior at this time scale. A power law probability distribution is a probability distribution in which the probability of an event decays as a negative power of the event. The distribution function is characterized by a scaling exponent. Distributions of returns typically exhibit power law decay in the tail of the distribution.
Here, we want to understand how the exact nature of power law behavior depends on the time lag between price observations.
We investigate how the scaling exponent changes as a function of the time lag between price observations. Our results suggest that the probability of finding large price changes is underestimated by a Gaussian distribution and better quantified by a power law distribution, in line with a range of findings reported in the field of econophysics [ 30 — 42 ]. Previous findings for US markets have highlighted that stock returns may follow an inverse cubic law [ 31 ].
The analysis of different stock markets, such as the Warsaw Stock Exchange in Poland or the Australian Stock Exchange, has uncovered different power law regimes deviating from the inverse cubic law [ 38 , 39 ]. By selecting appropriate cutoff values in the distributions under analysis, stocks from the Mexican Stock Market index exhibit a power law decay close to an inverse cubic law [ 40 ]. Analogous results have also been observed when analysing daily returns in Chinese stock markets [ 46 , 47 ].
We consider here the tails of the positive component of the distributions obtained when analyzing all trading days present in our dataset. It remains unclear, however, whether these findings hold for subsets of the price time series in which extreme price movements occur. We investigate this change in behavior at a range of time scales and analyze whether we start to observe consistency with exponential decay.
Exponential decay has already been observed in daily returns of stocks from the National Stock Exchange in the Indian stock market [ 48 ]. We investigate whether the tails of the returns distributions are consistent with power law behavior or exponential decay using the Kolmogorov-Smirnov statistic, as described in the methods section. We first consider all trading days present in our dataset. At short time scales, we observe that the tails of most empirical distributions are consistent with power law behavior.
As we increase the time lag, the number of tails consistent with power law behavior decreases and we see an increase in the number of tails of returns distributions that are consistent with exponential decay. We depict here the overall number of tails, both for the positive and negative returns distributions, for the 25 components of the DJIA.
In this scenario, the number of tails consistent with power law decreases more sharply. At small time scales, the tail of most distributions is consistent with power law behavior. As we increase the time lag between price observations, we observe an increase in the number of tails consistent with exponential decay.
As we increase the stress level, we find that the number of tails consistent with power law behavior decreases even more sharply. Large changes in stock market prices can occur at a range of time scales, arising within minutes or developing across longer time scales. Our findings provide evidence that in different scenarios the scaling exponent of those distributions consistent with power law behavior increases with the time lag between price observations. As this time lag increases, we observe that the number of return distributions consistent with power law behavior decreases sharply.
At a time lag of roughly two hours, we also find an increase in the number of distributions which are consistent with exponential decay. Our results are consistent with the hypothesis that changes in stock market prices have different behaviors at different time scales. We observe that these results hold in different scenarios of the market, both when we consider all trading days, but also when restricting our analysis to scenarios with different stress levels.
We suggest that our analysis may provide further empirical insights for the development of models of market behavior. To check the consistency of the tails of observed empirical distributions with power law behavior, we follow the procedure proposed in Ref.
We measure distances between distribution using the Kolmogorov-Smirnov statistic KS statistic :. We determine the lower bound x min by choosing the value that minimizes the distance between the empirical distribution and the fitted distribution as measured by the KS statistic. We construct the empirical tails choosing a bin size such that we have 1, data points in each tail.
We then compare the KS statistic observed for the empirical data when compared to a fitted power law distribution with the KS statistic obtained for the synthetic data when compared to a fitted power law distribution. We obtain a p -value by counting the number of times that the synthetic KS statistic is larger than the empirical KS statistic. We generate 1, synthetic data sets and make the conservative choice of accepting our hypothesis of consistency with power law behavior if the p -value is larger than 0.
To determine whether the distribution is consistent with exponential decay, we perform a parallel analysis fitting the data to an exponential distribution instead of a power law probability distribution. We then generate synthetic data from the fitted distribution in the same manner as previously described.
We evaluate whether our data are consistent with exponential decay by comparing the empirical data to the synthetic data using KS statistics as described above. National Center for Biotechnology Information , U. PLoS One. Published online Sep 1. Eugene Stanley , 3 and Tobias Preis 2. Eugene Stanley.
Yanguang Chen, Editor. Author information Article notes Copyright and License information Disclaimer. Competing Interests: The authors have declared that no competing interests exist. Received May 19; Accepted Jul This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.
This article has been cited by other articles in PMC.
Abstract Being able to quantify the probability of large price changes in stock markets is of crucial importance in understanding financial crises that affect the lives of people worldwide. Introduction Complex movements in stock market prices affect the personal fortunes of people around the globe [ 1 — 5 ]. Open in a separate window. Fig 1.
Components of the DJIA. Fig 2. Empirical distribution of normalised returns for American Express. Fig 3.
Fig 4. Consistency of empirical returns distributions with power law and exponential decay.