- Daily volatility of energy prices is similar to yearly volatility for stock markets
- How do you generate accurate models with spikes that are important to model correctly?
- Electricity markets are also heavy tailed (I think here they use Cauchy)
- Measuring heavy-tailed processes often result in instability

- Problem is non-stationary
- Leverage the fact that quantiles are not impacted by rare, high/low valued events
- “The trailing median process (TMP), which is a thin-tailed, symmetric, and a slow varying process with very low volatility, reverts towards the mean in the long term. The long-term mean f the TMP, /mu, is the long-term ‘average’ of the price process itself.” They refer to it as the long term mean
- “The law of large numbers is not applicable in a heavy-tailed environment and hence empirical averages have little statistical significance.”
- A common way to deal with very spiky data that they have here is by two separate: normal and jump, processes (jump diffusion model?), but it turns out that still doesn’t work well here
- This is because the tail follows a power, instead of exponential law (the def of heavy-tailed processes).
- Variance is not (effectively?) finite. This can be demonstrated by plotting the sample-variance based on the amount of data. It does not converge with tens of thousands of samples
- There are other reasons why jump-diffusion models aren’t so good for this task
- Because of high-variance, models that many DOFS and have good fit on training probably over-fit
- Even the mean (first moment – variance is second moment), is not well defined either; it also does not converge even with very large data sets. This means that the law of large numbers also does not apply. Sample means don’t tell anything about what should be expected in the future
- Thats a pretty wild environment to be making decisions in

- “In a heavy-tailed environment, higher order statistics such as the mean and the variance are unreliable. The only robust and verifiable statement we can make is that of the 0-th order statistics. For example, we can say something like the following: the probability that the electricity price in 4pm is greater than the electricity price at 2am is 80%. How much greater, though, is difficult to quantify both in the expectation or in the median.”
- Their model is based on the evolution of the median of price
- Martingale is used, which makes the error tiny (the average difference between the last and current means/curr mean)
- Use a window of 30 days to estimate median (breaking it down more or less doesn’t help much)
- Goal is to build predictions for 1 hour ahead (I think that is when bids need to go in)
- There is good empirical and theoretical backing as to why cauchy is a good idea here (based on an “estimator for the tail index”)
- Data seems to fit model very well, although they point out that errors in estimates compound for predictions far into the future

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