Jurgen Klopp and Harold Edwin Hurst

Klopp said "Well, I know that the longer a run is going, the more likely it is that it ends...". This can only be analysed using the Hurst Index

Ahead of today’s game at Manchester City (my original plan was to go to Manchester City fan Baada’s house and watch this, but then this happened! ), Jurgen Klopp has said this:

https://twitter.com/neiljonesgoal/status/1728043458974175451

(Ok it appears tweets don’t embed well in Substack. He’s said "Well, I know that the longer a run is going, the more likely it is that it ends...")

I’m thinking of the statistical validity of this quote. Does the likelihood of a run ending actually depend upon how long it is going for?

My initial answer was “no”. And then I realised that it depends on what kind of a process it is - whether it is momentum-following, or mean-reverting, or a random walk. And this takes me back to British hydrologist Harold Edwin Hurst, who I have written about extensively.

Quoting from my own article in Mint:

If a quantity were to follow a random pattern, the range of values it can take (defined as maximum minus minimum) over N time periods (what the time period is doesn’t matter) is proportional to the standard deviation of values, and proportional to the square root of N. Putting it differently, the ratio of the range of prices to the standard deviation of prices (which is known as rescaled range, represented as R/S, with R for Range and S for Standard Deviation) is proportional to the square root of the number of time periods.

If the time series has positive long-range dependence (or an upward move today more likely to cause an upward rather than downward move tomorrow), the potential range it can achieve in N time periods is going to be higher than that of a random walk (this is not hard to see. If an up movement results in further up movements, it is likely to go much farther from the origin). On the other hand, if the series has negative long-range time dependence (an example of this is a mean reverting series where any movements away from normal are likely to cause a bounce-back), then the potential range the series can achieve in N time periods is much less than that of a random walk

Ok that is a mouthful (and it confirms what my wife used to tell me - that the writing I got paid best for is also my worst and driest writing). Basically, the idea is - does a process have positive or negative or zero “serial correlation”? If something goes “up" today, is it more or less likely to go “up” tomorrow?

Standard stock market (and other asset price) models assume independence - that what happens today has no bearing on what happens tomorrow. The entire edifice of quantitative finance is built on this assumption. This independence is quantified in a H index of 1/2.

When a series is modelled with a H index strictly greater than 1/2 (such as the flooding of the Nile), then it means it is serially positively correlated. A good day today is going to make a good day tomorrow more likely. And this dependence holds over the longer term. Similarly, when the H index is less than 1/2, there is mean-reverting behaviour. A good day today is going to make a bad day tomorrow more likely.

Coming back to Klopp’s comment - what is the H index of football results? I mean, I have the data and can calculate, but right now NED (and I have other priorities in life). However, that is the only way to comment on Klopp’s comment. If the H index of home wins in the Premier League is < 1/2, he comment holds (that “the longer the run has gone for, the more likely it will end”). If the H index is strictly 1/2, then the length of the run doesn’t matter.

Maybe some day, soon, I’ll do this analysis.