# Skewness, kurtosis and normality of a time series

I have a sample size of $21$ with $496$ observations.Can I presume an approximately normal distribution,and use a $t$-test to compare the difference in means, and difference in various financial features (Financial data daily prices)?

In addition, sometimes I get insignificant differences on the daily basis, however, if I annualize the mean and the standard error, it becomes highly significant, like a difference between two means per day is $0.10%$ at the end of the year using this formula it will be extremely huge $((1+R)^{240}))-1$, any converting the standard deviation by the formula $\sqrt{240}* SD_{daily}$.

First it was a typo yes, the difference in the daily return is 0.10%, then the data will be auto-correlated for sure, since the market regulations allows the price to fluctuate by 2% only per trading session so the pattern will be there, what I am doing is Mainly constructing indexes in the market, grouping the 21 stocks into two groups, 16 stocks and 5, and then I will be comparing the indexes returns, Sharpe ratio etc, so when I'm using t-test to check statistical difference as this formula (x1-x2) difference in means / SQRt(SE^2+SE^2) the results were not statistically significant but financially 0.10% per day will be at the end of the year 15-17% difference which is MASSIVE!!!!!
and I provided the statistics for the daily data Kurtosis: the whole market (7.392221073) first group (6.344474009) second group (14.0065786) Skewness the whole market ,(0.930473559), first group (1.173843213), second group(1.325395262). So I'm actually taking the whole population which is only 21 firms in the emergent market, so in my research introduction chapter I have to talk about this issue so do you think I have to say by the central limit theorem, I have more than 20 sample size, and I would use t-test for comparing, if not it is not acceptable, tell me any other test for comparing the differences between the means and in details please.

Annual stats
market        set1          set2
Average Annual Return   0.182450075   0.344723515   0.149800603
Standard Deviation      0.134419078   0.202235211   0.138592387

per session stats      market   set1     set2
Mean                    0.10%   0.18%   0.08%
Standard Deviation      1.05%   1.57%   1.08%


the conversion done by assuming 165.33 session per annum

• I divided the sample into three groups: Kurtosis: first group (7.392221073) second group (6.344474009) third group (14.0065786) Skewness first group,(0.930473559), second group (1.173843213), third group(1.325395262) Aug 6, 2012 at 17:37
• Hi @James, welcome to the site. I tried to edit your question a little, make sure it still says what you want. Re annualizing the "standard error", do you mean the standard deviation? What exactly does it mean that you have a sample size of 21, but 496 observations? Might this be something like 21 stocks, w/ approximately 26 daily close values for each? Are your data clustered / nested in some way (eg, observations of the same stocks)? Is this time-series (are any observations ordered in time)? Re the presumption of normality, do you mean your sample data, or the sampling dist via the CLT? Aug 6, 2012 at 17:45
• 21 stocks, Observations: The average price per trading session of each stock for 496 trading sessions (3 sessions a week), so 3 years (496 trading days). standard error = standard deviation / SQRT(N) so annualizing any of them would do the task to get the other, Aug 6, 2012 at 17:56
• So columns are 21 stocks, rows are 496 trading days, so in each trading day I have the average price for each stock Aug 6, 2012 at 18:00
• I just noticed that you wrote 0.10 as daily mean differential ? do you mean 10% ? this is already huge for daily returns (this may be the cause of innacuracy of the scaling formula). Or did you mean 10 basis points = 0.10 %= 0.001. It might be a good idea to check your measurement units.
– JDav
Aug 6, 2012 at 20:05

I'm afraid I don't know much about the analysis of financial features, so I'm limited in what I can tell you, but I can say some general things. (@JDav sounds pretty authoritative, but I'm not qualified to evaluate it.)

• In general, it doesn't matter if your data are normally distributed, only if your residuals are (which is explained here) this even true for a t-test.
• No amount of data will turn a non-normal sample into a normally distributed one.
• With respect to the validity of the t-test (more specifically its p-values), the question is whether the sampling distributions of the means, and the sampling distribution of their difference, is normally distributed. With so much data, the central limit theorem is likely to cover you unless you have very heavy tails.
• Given that your data are clustered / ordered in time, I doubt a t-test is appropriate. You almost certainly need a multilevel / time-series model.
• In my answer I forgot to asses the normality issue as I was surprised by the significant difference between daily and scaled returns. But I completly agree with gung's answer, the CLT is likely to apply in your case.
– JDav
Aug 6, 2012 at 20:03
• Of course I may be wrong but one of the core principles of this web site is to give an answer only if we really believe that we have one, otherwise, "answers" are just usefull speculations that might help to come up with an answer.
– JDav
Aug 6, 2012 at 20:17
• @JDav, that is one of the core principles of this website. I know a little about time-series & related topics, but try to stay away from them, & let more expert CV contributors address those issues. I frankly know next to nothing about financial modeling, & admit that openly at the beginning of this answer. I do, however, know a little about other statistical topics; I stand behind what I've written here, eg, that if the data are distributed w/ fat tails (say, as t w/ 3df) then no amount of data will make the sample converge to normallity, but that w/ enough the sampling dist of the mean will. Aug 6, 2012 at 20:28
• The time series nature of the data means that the ordinary t test is not appropriate as gung pointed out. The t test is assuming indpenednet observations whereas you may have autocorrelation that affects the variance of the estimate. Sometime correlation structure can be removed by taking paired differences in which case the paired t test may be okay. For example i ahd a data set that compared daily temperature in NYC and Washington DC. Both series are highly seasonal but the daily paired difference in temperature was not. Aug 7, 2012 at 22:09

Your formula( scaling by $\sqrt{240}$) works as long as R is close to zero. Both daily and yearly t-stat should be almost equal on that case, if not, that might be an indication of the poor approximation of the scaling formula (remember that it is based on a Taylor expansion at r=0).

Moreover, even if r is close to zero this formula assumes no autocorrelation, which is that your prices today are not correlated with your prices yesterday. Dependng on your data that could be a heavy assumption. To keep your analysis simple, you may try to stay on a daily basis horizon, this will avoid the poor approximation and the non autocorrelation issues.

Nevertheless it's also true that stock returns are well known for being better explained by non autocorrelated process. But this depends on your data and can be easily verified by calculating a regression $r_t=a+\rho r_{t-1}+u_t$ where $\rho$ would be significant in the presence of correlation (of course this can be generalized to higher orders).

• Another thing the range, the maximum, and the minimum are huge compared to returns per session for instance, the market return per session (mean) 0.10%, the range, maximum, minimum are 12.5%, 8%, -4.5% Aug 7, 2012 at 4:42
• I disagree with this method of scaling (see: papers.ssrn.com/sol3/papers.cfm?abstract_id=1586656). However, I agree that if he is looking at daily data, he should perform statistical tests on the daily data. Some financial time series (not stocks, but interest rates) will exhibit autocorrelation. In which case, it is often common for t-stats to be calculated using the Newey-West standard errors.
– John
Aug 7, 2012 at 18:10