I am analyzing stock returns in a time series. Some values are negative and other positive. While skewness in a normal distribution is = 0, the normal seems to be the most suitable because it allows negative values. I end up with a positively skewed distribution. Lognormals and betha distribution cannot be considered because they do not allow for negative values. Is it proper to depart from a normal distribution under these circumstances, and are the results valid?

If what you are plotting is the percent return for a each time it will naturally skewed. A +50% return one year is balanced by a -33% return in another year, not -50%. Another example: while it's possible to have a +200% return, it's impossible to have a -200% return. So if what you're plotting is return for each time it will be skewed. It would be better to plot the logarithms of the "growth factors" - that is, the number you multiply the previous periods value by to get the next period's value. For example, for 0% return in a given time use log(1), for a +10% return use log(1.1), and for a -10% return use log(0.9). This technique will result in the +100% return being properly "balanced" by a -50% return: i.e. log(2) versus log(0.5).

Hope this helps.

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Originally Posted by ebaines

If what you are plotting is the percent return for a each time period it will naturally skewed. A +50% return one year is balanced by a -33% return in another year, not -50%. Another example: while it's possible to have a +200% return, it's impossible to have a -200% return. So if what you're plotting is return for each time period it will be skewed. It would be better to plot the logarithms of the "growth factors" - that is, the number you multiply the previous periods value by to get the next period's value. For example, for 0% return in a given time period use log(1), for a +10% return use log(1.1), and for a -10% return use log(0.9). This technique will result in the +100% return being properly "balanced" by a -50% return: i.e. log(2) versus log(0.5).

Hope this helps.

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Thank you kindly for you answer. I think I did not pose the question correctly. What I really want to know is, whether the results generated by a "skewed" distribution which started as a normal distribution are valid results, even if it is not a normal distribution anymore. I am saying this, because I don't know of any other distribution I can use to plot different stock returns, which would allow negative returns. What's your opinion?

I am getting out of my comfort zone here, but if I understand you correctly the problem is that you are trying to analyze a skewed data set using techniques intended for a normal distribution - do I have that right? If so, then performing a suitable transformation on your data can help normalize it. I suggested a logarithmic transformation, which I still think should work OK. Here are a couple of web pages that talk about transformations of data sets to get normal distribution behavior, specifically using something called Box-Cox transformations:

1.3.5.11. Measures of Skewness and Kurtosis
1.3.3.6. Box-Cox Normality Plot