# Question on statistics, kurtosis and the financial crisis **URL:** \ **Category:** Factual Questions\ **Created:** [August 6, 2009, 12:29pm UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520 "2009-08-06T12:29:22Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![Saffer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saffer/32/6440_2.png) [@Saffer](https://boards.straightdope.com/u/Saffer)\ **Post date:** [August 6, 2009, 12:29pm UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520/1 "2009-08-06T12:29:22Z")
I am not a statistician but I do have some academic knowledge on the subject. Every now and then (on an exceptionally boring day) I open a new spreadsheet and generate some random numbers and then transform them using some algorithm. I have noticed that sometimes the set exhibits very high kurtosis (sometimes more than 15000) and when I check to see how many numbers fall within x standard deviations from the mean, the results are very non-normal as expected. I have often heard that very high kurtosis is a common complaint in asset pricing so my question is, is there any systematic theoretically accurate way of correcting/compensating for extreme kurtosis so that one can construct reasonable confidence intervals? If not, then doesn’t that make a lot of asset pricing (especially CDOs) a bit of a scam? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 6, 2009, 3:25pm UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520/2 "2009-08-06T15:25:59Z")
Pricing these days is based on the [GARCH model](http://en.wikipedia.org/wiki/GARCH#GARCH), which allows for much fatter tails than a standard ARMA model, or something similar. ---
**Author:** ![Saffer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saffer/32/6440_2.png) [@Saffer](https://boards.straightdope.com/u/Saffer)\ **Post date:** [August 6, 2009, 7:43pm UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520/3 "2009-08-06T19:43:17Z")
Thanks. I have actually heard of GARCH but I haven’t gone into too much detail and I wasn’t aware it is so popular. Does anyone know of any methods that have been developed specifically to deal with excess kurtosis? What I am wondering is if there is any test statistic that uses the kurtosis, the degrees of freedom and other parameters to perform a valid hypothesis test, perhaps using some special distribution as a model. Everything I have come across works on the assumption that the whatever you are looking is normally distributed or at least the error terms are normal (which I believe GARCH does also but not 100% sure). So is there any mathematical use for the kurtosis measure other than to say “look, it has excess kurtosis!”? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 6, 2009, 8:29pm UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520/4 "2009-08-06T20:29:35Z")
You can’t just look at the kurtosis of an asset over a given time period because that doesn’t tell you everything that’s going on. Consider the returns on the S&P 500 for the period starting 1/1/1999 and ending 12/31/2007. Simply calculating the kurtosis would lead you to believe that they’re not normally distributed. On the other hand, if you plot them, it’s painfully clear that the volatility is non-constant, and that you need a model that can handle that, not just the excess kurtosis. ---
**Author:** ![bleach](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bleach/32/3106_2.png) [@bleach](https://boards.straightdope.com/u/bleach)\ **Post date:** [August 7, 2009, 1:06am UTC](https://boards.straightdope.com/t/question-on-statistics-kurtosis-and-the-financial-crisis/505520/5 "2009-08-07T01:06:09Z")
Couple of questions: For your random numbers, are you simulating from a uniform distribution or a normal distribution or another distribution? You also said that you do a transformation. Is the transformation a linear or logarithmic transformation or another non-linear transformation? One wouldn’t expect to see a normal distribution in the raw data after any given non-linear transformation. Some families of transformations preserve kurtosis but many do not. A statistic like the mean has the central limit theorem machinery that asymptotically converges to a normal distribution but raw data may or may not be normally distributed given a set of transformations. Another set of questions is: what are you trying to test? Are you testing a kurtosis hypothesis? I’m not sure why one would want to consider a kurtosis test alone as it relates to pricing and markets, but yes, there are tests for kurtosis. However, let’s say that there’s excess kurtosis; are you asking whether that turns into some sort of arbitrage opportunity that one can exploit? Like **ultrafilter** said, in a dynamic system, a snapshot of a statistic isn’t particularly informative because so many other things (including summary statistics) are changing simultaneously. In statistical circles, there isn’t a huge clamor for kurtosis reporting. It’s a statistic, but there are lots of other statistics that are more easily interpreted for summarizing data.