Normality test

paolopiace · Post by **paolopiace** » Wed Apr 13, 2011 8:46 pm

Hello.

I did run the Jarque-Bera (JB) normality test on some equities.

In order to account for the last two recessions and market crash the JB test was performed on the sample of 2605 daily returns covering the last ~10 years.

The JB test was performed after having removed the returns bigger than 5 StDev, considered as outliers. For removal, the amplitude of the outliers was compared to the StDev of the daily returns (calculated as sqrt of the the unbiased variance respect to the mean).

One stock (DO) and one ETF (SMH) deserved the most attention:

SMH:

StDev = 0.39, Skewness = +0.142, Kurtosis (excess) = 2.237, JB=548; Outliers = 2 (Tails prob. = 2/2605 = 0.077%)

DO:

StDev = 0.39, Skewness = -0.031, Kurtosis (excess) = 1.271, JB=174; Outliers = 6 (Tails prob. = 6/2605 = 0.230%)

Although the JB test grossly fails on both, it fails more on SMH.

But that seems to contradict the outliers: SMH has way less outliers than DO.

So, I've hard time to interpret these results: Jarque-Bera tells that DO is more normal than SMH why the outliers tell otherwise.

I'm seeking a shed of light as well as opinons...

Thank You!

nodoodahs · Post by **nodoodahs** » Wed Apr 13, 2011 9:28 pm

[stuff about including the outliers and retesting]

Rethinking and restating my response.

Extreme edges of the distribution don't define deviations from normality, so we can't look at the counts of so-called "outliers" as a relative normality measurement.

SMH has a fatter middle than DO. 25% of the normal distribution is between +/-0.319 SD and those values are 35.9% for SMH and 28.1% for DO.

While the tails are fatter for DO, that's only at the extreme ends. 25% of the normal distribution is outside of +/-1.15 SD. That value is 19.0% for SMH and 20.3% for DO.

I did a chi-squared on 8 quantiles of 'normal' for SMH and DO, and SMH scored higher a.k.a. 'less normal.' Not familiar with the test you listed, but wouldn't be surprised if it's essentially based on some generalization of the chi-squared.

paolopiace · Post by **paolopiace** » Wed Apr 13, 2011 9:51 pm

Good point. Thanks.

Including the outliers the JB test tells the opposite than before: SMH is more normal than DO.

SMH: StDev = 0.40, Skewness = +0.278, Kurtosis (excess) = 2.775, JB=864

DO: StDev = 0.41, Skewness = -0.181, Kurtosis (excess) = 5.241, JB=2981

I then tend to conclude that both the JB test and the tail thickness (outliers) tell that SMH is more normal than DO.

Hence... more analysis to do.

paolopiace · Post by **paolopiace** » Wed Apr 13, 2011 9:57 pm

By the way...

If someone has strong substantiated opinions about Normality Tests, i.e. which is better performing and reliable, I'd be eager to know.

nodoodahs · Post by **nodoodahs** » Wed Apr 13, 2011 10:45 pm

Moderately strong and totally unsubstantiated opinion here. None are "better." Some may be more useful than others, depending upon the context.

Here's an example, I did a chi-square Vs. 'normal' with 8 quantiles and with 10 quantiles.

With 8 quantiles, SMH scored less 'normal' than DO.

With 10 quantiles, DO scored less 'normal' than SMH.

Same test, different quantile count. Is one better?

You ran J-B (I'm unfamiliar with it) with and without 8 data points out of 2 x 2605 data points and got two different results. Statisticians could (and would) argue all day about whether excluding outliers is appropriate, depending upon context and the outliers (and occasionally depending on whether they like the results with/without the outliers).

Lots of tests here http://en.wikipedia.org/wiki/Normality_test

paolopiace · Post by **paolopiace** » Wed Apr 13, 2011 11:13 pm

Nodoodahs, Thanks a bunch. Very appreciated.

I'll try to run the chi-square also.

Point is that, among the ~350 equities I tested, SMH came out with the thinner, most normal, tails while DO came out with the best JB score after outliers removal.

Essentially and for different tests, they behaved as the most "normal" among the 350 during the last 10 years.

Yes, I stepped into that wiki page weeks back. I then decided to stick to two tests: Jarque-Bera and Kolmogorov-Smirnov. But the implementation of the second is still in progress. Let's see what it will tell.

All the best...

P.S. I've just now seen the update on your posting of 12:28pm. Thanks for that. By the way, the JB essentially measures the asymptotic limit of the variance of skewness + variance of kurtosis. Hence, being the sum of two squares, it has a Chi-square distribution with 2 degrees of freedom.

aaron · Post by **aaron** » Sat Apr 23, 2011 10:13 pm

For most financial applications, you don't care about the unconditional Normality over the past ten years, you care about the conditional Normality of the return today, over some future interval. We know that equity volatility goes up and down, you are interpreting that heteroskedasticity as non-Normality.

A better test is to look at the distribution of returns divided by a measure of their volatility, implied volatility for example, or realized volatility over the three months prior to the return, or the errors after a GARCH fitting. You want your observations independent before you worry about their distribution.

As for the best test, it's always a direct test of what you care about. If you're pricing options, test option prices. If you want to know the probability of moves more than 3 standard deviations from the mean, count the number of moves more than 3 standard deviations from the mean.

nodoodahs · Post by **nodoodahs** » Sat Apr 23, 2011 11:07 pm

>look at the distribution of returns divided by a measure of their volatility, implied volatility for example, or realized volatility over the three months prior to the return

Worship

paolopiace · Post by **paolopiace** » Mon Apr 25, 2011 5:10 pm

Aaron, I see what you mean. Thank you.

Roughly speaking, I read that the measurement of Normality should be restricted among returns (hence their distribution) that are subject to the same level of volatility.

Though, please, let me know if I'm interpreting correctly your statement:

>> You want your observations independent before you worry about their distribution.

My interpretation: independence here refers to the volatilities. In case of implied volatilities, they can be assumed independent. In case of realized volatilities, they are independent if computed with non-overlapping series of returns.

In case of overlapping series, volatilities are highly correlated, hence dependent. Their dependency bias must be corrected as described by Hodges & Tompkins in the Volatility Cones; that's the only method I'm aware of.

Is that what you meant?

nodoodahs · Post by **nodoodahs** » Mon Apr 25, 2011 5:58 pm

BTW ...

SPY dailies decidedly non-normal after adjusting for trailing vol; removal of the heteroskedasticity makes it 'more nearly normal' but still flunks a Chi^2, generates 3.4x as many Z > 3 movements, still has a 'hump' between Z of 0 and 1.

Good side is, excess Kurt goes from 10.9 to 1.8 once returns are adjusted for trailing vol.

Used mean true range (average over 45 trades) divided by mean typical daily price (45 days) to generate trailing vol, but I guess any measure (stdev, etc) would end in about the same result, i.e. 'more nearly normal' but still 'provably abnormal'.

I do the like the 'removal' of the heteroskedasticity trick, though, thanks.