# What to do if no probability distribution accurately represents my data?

I am trying to find what distribution would fit logarithmic stock returns. I already did a Jarque Bera test with my data and got a p value smaller than .05

library("TTR")
library("tseries")
sign="^GSPC"
start=20150101
end=20160101
x <- getYahooData(sign, start = start, end = end, freq = "daily")
x$logret <- log(x$Close) - lag(log(x$Close)) x<-x[,6] jarque.bera.test(as.numeric(na.omit(x))) Jarque Bera Test data: as.numeric(na.omit(x)) X-squared = 637.01, df = 2, p-value < 2.2e-16  I also tried to look for some possible candidate solutions from How to determine which distribution fits my data best? with the Cullen and Frey graph library("fitdistrplus") descdist(as.numeric(na.omit(x)), discrete = FALSE)  So apparently no distribution is a good candidate for the data I am using. What is the step to follow? EDIT: Here is an histogram of my data • Could you post histogram or density plot of your data? What do you know about it? In most cases you do not find distributions by exhaustive search and testing but rather assume one that nicely approximate your data. Btw have you seen: finance.martinsewell.com/stylized-facts/scaling/Officer1972.pdf – Tim Sep 21 '16 at 14:30 • What is your goal? Why do you need to fit a distribution to your data? – Roland Sep 21 '16 at 14:38 • Given any dataset, no matter how large, there are infinitely many distributions that will fit it. The comments suggest your real problem is nothing like the question you have posted. It might be better to start over with a more suitable question. – whuber Sep 21 '16 at 15:36 • @whuber I don't know how to make goodness of fit for t distribution, that being said, that is not my main concern. The reason why I posted this is because I want to know what should I do if no distribution fits in my data. What do you mean by "there are infinitely many distributions that will fit"? How do I apply that on my data? – Rafael Martínez Sep 21 '16 at 15:47 • Trying to fit some distribution to your data sounds like an unconstructive approach to whatever it is you're trying to do. Given any dataset, anyone can find a distribution that will fit it perfectly. But the exercise is pointless. – whuber Sep 21 '16 at 15:50 ## 2 Answers Your Cullen and Frey graph clearly suggests a distribution with low skewness (consistent with symmetry, for example) and high kurtosis. The diagram unfortunately doesn't tell you any, but that doesn't mean there aren't any! There's an infinite number of distributions with those properties, but if you're after a distribution in the Pearson family (which is what using such a graph would tend to suggest you're after) that would suggest a t-distribution -- the point is very close to the t-line (Pearson type VII; the green half-line running from the$N$to the right of the plot). Note that this plot is flipped around compared to the Cullen and Frey plot; your plot's t-line runs down the page. (Specifically, this would be a t distribution with three parameters - df, location and scale), though among the Pearson family it could as easily be a Pearson type IV (one close to t in appearance but not actually perfectly symmetric), As it happens, using$t$distributions to model log-returns is very common. [However there are often difficulties in estimating degrees of freedom, location and scale simultaneously.] The suggestion from the kurtosis value is something in the rough ballpark of 4.7 df but simulation suggests that sampling from t-distributions (even with n over 6000) will frequently turn up a kurtosis of about that size for degrees of freedom from somewhat below 6 down to well below 4$^\dagger$; 5 is quite consistent with it, as is 3.5. Note however, that simple distributions are models (i.e. approximate descriptions) not exact. If you have a large enough sample size (and a suitable test*) you'll always reject some simple distributional model even if it's a really good approximation and entirely suitable as a model. With larger and larger sample sizes you can detect what are essentially trivial differences from the model. To actually assess whether your model is reasonably accurate I would suggest neither a goodness of fit test nor a histogram (though if you add a lot more bins it might not be so bad) - I'd suggest looking at a Q-Q plot or a P-P plot instead.$\dagger\$ even though 4df implies infinite population kurtosis, the modal sample kurtosis for 4 df and n=6000 is still below 10)

* The Jarque Bera would not identify deviations in distribution that aren't reflected in skewness and kurtosis. Very different distributions can have the same skewness and kurtosis. It lacks power against alternatives that don't differ on at least one of the two.

This is a widely studied problem in quantitative finance. You've selected and posted only one year worth of data. If your time series was extended back to 2007, you would see ever greater divergence in your chart with the inclusion of the market crashes back then.

Returns are well known not to be normal, log normal or even t-distributed. All three distributions show significant lack of fit vis-a-vis returns. There's the famous story, perhaps told best by David Hand in his book The Improbability Principle (p. 147), of the Goldman-Sachs CFO who testified in Congress after 2007 that, based on the normal distribution, the recent financial shocks were "25-standard-deviation events, several days in a row." The fact is that, based on the normal, there isn't enough time in the history of the universe to capture a 1 in 25-sigma event probability. Thus, it's no surprise that the Jacques-Bera would reject an assumption of normality...but then, the J-B rejects normality almost every time it's applied.

Hand goes on to note, "If the distribution of market fluctuations isn't normal, it must have some other shape." His suggestion of a better fit to stock returns is the Cauchy distribution. And he provides a table comparing the normal and the Cauchy in terms of the probabilities of event moves. Based on this table, it's evident that we should expect rare events to occur with much greater frequency:

As @glen_b notes, Q-Q plots are a sensible way to examine and compare fit or the lack of it. Another approach that is quite useful with extreme valued information is to estimate a tail index. The literature concerned with extreme value theory is quite large and estimands of this index include Hill and Pickand's methods -- both computationally intensive. A more straightforward heuristic is to leverage Gabaix's OLS-based approach using the log of the ranks of the data as described in his paper Rank - 1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents. (An ungated copy can be found here: http://www.eco.uc3m.es/temp/jbes.2009.06157.pdf). Gabaix contends that his approach is mathematically well-founded but it's not without controversy. For instance, Shalizi argues in his presentation So You Think You Have a Power Law Do You? (an ungated copy can be found here http://www.stat.cmu.edu/~cshalizi/2010-10-18-Meetup.pdf), that OLS estimators of tail exponents are "bad practice." I won't take sides on this, you be the judge.

Once one has estimated a tail exponent, there's a bit of a cheat sheet on this Wiki page about Tweedie Distributions. Look under "Examples," where the magnitude of the index is aligned with possible explanatory, extreme valued distributions (https://en.wikipedia.org/wiki/Tweedie_distribution).

One individual who has thought and continues to think deeply about the implications of extreme market moves is Didier Sornette, now Chair of the Financial Crisis Observatory at ETHZ. His 2004 book Why Stock Markets Crash: Critical Events in Complex Financial Systems is a superb overview on the manifold complexities in analyzing stock returns. In addition, his ETHZ website has many, many more papers examining a huge many other aspects of these problems. The sophistication of Sornette's views on markets, bubbles and risk more than repays the effort involved in understanding them.

Other useful observers, practitioners and developers of EVT (extreme value theory) include Paul Embrechts, also at ETHZ, whose 1996 book Modeling Extreme Events remains a classic and the bible in the field. Then, there's always Nassim Taleb whose free book Silent Risk can be downloaded from his website: https://fernandonogueiracosta.files.wordpress.com/2014/07/taleb-nassim-silent-risk.pdf. Taleb can be a bit of a blowhard but his views, while always controversial, are widely cited.

To summarize and answer your query, it is definitely not the case that "no distribution summarizes your data." You've been given a variety of tools and resources to tackle this problem in these responses. So, go out and pick a distribution, then motivate it.

• +1. I'm curious, though, what a "5-sigma" excursion for a Cauchy distribution might be, since "sigma" for a Cauchy distribution is infinite! (Evidently the table assumes "sigma" is unity--but what that means for the Cauchy distribution is unclear.) – whuber Sep 21 '16 at 21:21
• @whuber Good point. I would be second-guessing David Hand if I attempted to answer you. Having corresponded with him, he is amenable to email. Why not ask him? – Mike Hunter Sep 21 '16 at 21:27