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 sequntially 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 basically only finance.

Read!

BaSø94
Barndorff-Nielsen, O. E., & Sørensen, M. (1994) A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes. International Statistical Review / Revue Internationale de Statistique, 62(1), 133–165. DOI.
BiSø95
Bibby, B. M., & Sørensen, M. (1995) Martingale Estimation Functions for Discretely Observed Diffusion Processes. Bernoulli, 1(1/2), 17–39. DOI.
DuPo15
Duembgen, M., & Podolskij, M. (2015) High-frequency asymptotics for path-dependent functionals of Itô semimartingales. Stochastic Processes and Their Applications, 125(4), 1195–1217. DOI.
Feig76
Feigin, P. D.(1976) Maximum Likelihood Estimation for Continuous-Time Stochastic Processes. Advances in Applied Probability, 8(4), 712–736. DOI.
HePo14
Heinrich, C., & Podolskij, M. (2014) On spectral distribution of high dimensional covariation matrices. arXiv:1410.6764 [Math].
HeSe10
Heyde, C. C., & Seneta, E. (2010) Estimation Theory for Growth and Immigration Rates in a Multiplicative Process. In R. Maller, I. Basawa, P. Hall, & E. Seneta (Eds.), Selected Works of C.C. Heyde (pp. 214–235). Springer New York
Jaco97
Jacod, J. (1997) On continuous conditional Gaussian martingales and stable convergence in law. In J. Azéma, M. Yor, & M. Emery (Eds.), Séminaire de Probabilités XXXI (pp. 232–246). Springer Berlin Heidelberg
JaPV10
Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. The Annals of Statistics, 38(3), 1478–1545. DOI.
PoVe10
Podolskij, M., & Vetter, M. (2010) Understanding limit theorems for semimartingales: a short survey: Limit theorems for semimartingales. Statistica Neerlandica, 64(3), 329–351. DOI.