The Living Thing / Notebooks :

High frequency time series estimation

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.