Afternoon allI have a question related to "weather returns". I'm using the modeldS=a(m-S)dt+v dW .....(1)to model the weather. I want to look at the "volatility" of the weather. Obviously I can't use the standard definitions of dS/S or ln(S+/S), since in this case they are meaningless with respect to the model (1) - not to mention 0 and negative values. So I have taken the transform X=exp(S) which givesdX/X = a(m+0.5 v^2-lnX)dt+vdW=> Var(dX/X)=v^2 dtThe problem I am having is that for the Heathrow temperature from 1961-2003 I have a huge volatility - it has numerous outliers with v approximately 900. The problem is caused by huge outliers due to swings in the temperature. For instance in December 1961 the weather temp from one day to the next was -1 to +7 - this leads to a "return" on that day of over 90,000. This was the worst day but there have been a lot of similar ones.How should I deal with this? If I strip out the outliers (>3*v) then I end up stripping out a hell of a lot of days but get a stable distribution? This seems a bit wrong because this is indicative of a jump model which as far as I know is not what the weather is?!?!Any suggestions would be greatly appreciated, cheersPablos

Don't strip out the outliers, unless you think they represent data errors.The trick is to choose a transformation that leads to a reasonably Gaussian distribution, or in your case, distribution of changes. Exp didn't work. My guess is you don't need any transformation at all, that the chance of a move from -1 to +2 is about the same as a move from 10 to 13 or 27 to 30. But if the data appear otherwise, you can transform.

Thanks AaronThe reason i was looking at a transform is becuase of the occurence of near zero temperatures in the winter. When the weather is zero the return is infinite and so I get a huge variance as a result of 0 temperatures. Also negative temperatures mean I can't use log(S(t+1)/S(t)).... However these are not extreme events and should undoubtably be included and should not be removed.I suppose that in general I was wondering if there was a way of looking at the "volatility" of a process that mean reverts around zero as the "returns" will exhibit this behaviour in a way similar to what I have done.... There always seem to be huge spikes in the "returns" (caused by "normal data") no matter how I look at the data - how do people generally deal with this?Many thanks Paul.PS I have included a spread sheet of the data I am looking at just to try and illustrate what I am saying...

Would using degrees Kelvin rather than degrees Celsius solve your problem?

whats wrong withS(t+1) - S(t)?

I've used S(t+1)-S(t), but all that gives me is a distribution of the size of the temperature change. It's the same thing as knowing that the price changes by £2, it means nothing unless you know what the change means relatively. A £2 change is more significant if the price was £10 as opposed to £1 000 000.I really need a handle on the volatility of the temperature, but I am having problems as I've outlined.

You are right about money, but wrong about temperature. The ability to turn $0 into $0.01 will make you infinitely rich, the ability to turn $100 into $100.01 is a lousy basis point. But with temperature, the change from one degree to another is more or less the same. If anything, the changes are more significant in the tails. Also the zero point is completely arbitrary.Start with the untransformed data, that is just the changes themselves. If they are roughly bell-shaped, most standard statistical techniques should work okay. If they deviate strongly then transform to make them more Gaussian. But remember, that's only for analytic convenience.

Just for an example, I downloaded some daily temperature data for Phoenix (I just happened to find it on the web). The daily changes pass standard tests for Normality. The series did exhibit strong mean reversion, the average change was greater by 1 degree for each 5 degree increase in temperature. I'm sure there are other effects as well, such as (literal) seasonality. So my advice is to forget the transformations and go into time series analysis.