Continued musings on the time-series analysis of the Central England Temperature series have lead me to look at processes for the variance of the temperature. I found seasonal heteroskedasticity, which i don't regard as suprising, and autoregressive heteroskedasticity — i.e. GARCH — which I do think is kind of interesting. It means that the climatic process that generates our weather can undergo “bursts” of volatility that persist for a while and then ebb away. So an anomalously hot month is just as likely to be followed by either an another anomalously hot month or an anomalously cold month but less likely to be followed by a normal month. Perhaps this is what makes the weather so confusing!

Anyway, p-Value laden statistical analysis is available at the SSRN

I have written extensively on my blog about the relationship between the returns of assets and their volatility. It is a point of interest. Commentators on financial television frequently discuss the performance of the VIX and compare it to the performance of the market. This common usage leads us to believe that the VIX is a factor we should pay a lot of attention to.

But empirically we do not see this. What we see is that the daily returns of the market, for which I'll take the returns of the SPY ETF, and the returns of the VIX, for which I'll take the returns of the VXX ETF are stabily and extremely negatively correlated. The chart below illustrates a linear regression of the returns of VXX onto the returns of SPY for several recent years of daily data. This negative relationship is extremely clear and strong (the R² is some 69%, which is a big value for the returns of different assets in finance — it's equivalent to a correlation of −83%). 

From a statistical point of view: when the market goes up the VIX goes down and when the market goes down the VIX goes up. Now these assets are not identical and their long-term trends do differ, hopefully the market ultimately drifts upwards whereas the VIX mean-reverts about a “typical” value, but on a daily basis they really disagree completely on direction in a very predictable manner.

In finance we are interested in assets that diversify our portfolio. This means that when add an asset to our portfolio the consequence ought to be that the net variance of the whole portfolio decreases. In their book on active portfolio management, Grinold & Kahn give the following formula for the risk of a portfolio of N assets all of which have a common pairwise correlation coefficient, ρ.  

I recently wrote up, and submitted to the Social Science Research Network, a study of the Central England Temperature series that I did a couple of years ago and recently updated. You can access the paper at the SSRN here: A Seasonal Autoregressive Model for the Central England Temperature Series.

What I do in this analysis is build a seasonal autoregressive model for what is the longest time series of directly measured temperatures available for analysis. This model includes a linear trend component. I fit the model for both the pre-Industrial Revolution Period (1659 to 1849) and the Post-Industrial Revolution period (1949 to 1999). In the former period no trend is apparent, in the latter period I find a warming trend of 1 °C/century. This result has a borderline statistical significance (less than 3σ). I use both models to forecast the temperatures for the 21st. Century and find that the older, trend free, model does a slightly better job out-of-sample than the newer model with the trend.