# High frequency time series estimation

June 12, 2016 — December 2, 2015

a.k.a. “Fancy ARIMA”.

Classically, you estimate statistics from many i.i.d. realisations from a presumed generating process.

What if your data are realisations of sequentially dependent time series? How do you estimate parameters from a single time series realisation?

By being a flashy quant!

Bonus points: How do you do this with *many* time series, whose parameters themselves have a distribution you wish to estimate?

See Mark Podolskij who explains “high frequency asymptotics” well. I think that the original framework is due to Jacod. (i.e. when you don’t have an asymptotic limit in number of data points, but in how densely you sample a single time series.)

This feels contrived for me, but it is probably interesting if you are not working with a multivariate Brownian motion, but a rather general Lévy process or something with interesting jumps AND continuous movement, and can sample with arbitrary density but not arbitrarily long. AFAICT this is little outside finance.

## 1 References

*International Statistical Review / Revue Internationale de Statistique*.

*Bernoulli*.

*Stochastic Processes and Their Applications*.

*Advances in Applied Probability*.

*arXiv:1410.6764 [Math]*.

*Selected Works of C.C. Heyde*. Selected Works in Probability and Statistics.

*Séminaire de Probabilités XXXI*. Lecture Notes in Mathematics 1655.

*The Annals of Statistics*.

*Statistica Neerlandica*.