Normality Assumptions and Classical Models for Tick Data

# Applying models with normality assumption on tick data?


# Applying models with normality assumption on tick data?












Beginner question. Having read a couple of papers and book chapters on high-frequency data forecasting, I'm surprised (and confused) that the same time series techniques can be applied to high-frequency/ tick data than for lower frequencies.

High-frequency data seems to clearly not follow a normal distribution given that prices change in discrete steps, have a minimal tick size, lag-1 autocorrelation, etc. Thus how can researchers apply techniques on it which have strong normality assumptions? (e.g. analysing the mid-price evolution and signed volume of transactions using vector autoregression of Hasbrouck).

For example [1] describes an autoregressive (VAR) quote-revision model which relates mid-price changes to signed trade size. Clearly, the difference in mid-prices are discrete and I would say ressembles count data.

[1] Measuring the Information of Stock Trades Joel Hasbrouck, Quantitative Finance, 1991, Journal of Finance pdf

## Answer by SRKX (score 4)

https://quant.stackexchange.com/a/2190

I'm trying to formulate an answer for latter users of the site and, hopefully to lure some more experienced high frequency traders to further react to my post.

First of all, I'd notice that the paper you mentioned was published in 1991. I believe high-frequency trading has evolved a lot since then, and that nowadays HF strategies are more evolved.

From what I know, high-frequency trading involves many different kinds of strategies, some of which are not related to normality assumptions. For example, if you want to trade arbitragist strategies, you will have to trade at high frequency in order to catch the arbitrage opportunity (otherwise, someone quicker might have taken advantage already, and the opportunity is gone), however this strategy clearly doesn't rely on the normality assumption. Trend following at high-frequency (which was the subject of my master thesis) doesn't rely on probabilistic assumption either, its more a technical or behavioral strategy.

From a more general point of view, you could also discuss the validity of the normal assumption, even at a higher frequency as Sornette quite brilliantly shows in his book. There is a quote from the Origin of Wealth that I quite fancy relating finance research always assuming unrealistic assumptions in finance "bullshit in, bullshit out".

As a matter of fact, you will have to add some noise to your model in order to make it realistic, but this noise might well be part of a framework which makes the overall distribution not really normal, or at least with a volatility that significantly evolve though time (Asymmetric GARCH for example).

Hence, I seriously doubt that any profitable strategy (on the long-term) really rely on pure normality assumptions.

I'd be more than happy to be proven wrong.

## Answer by eillasti (score 1)

https://quant.stackexchange.com/a/59552

Because there is no (publicly accessible?) viable alternative and classical models are good enough.

There are models that explicitly address the irregularities of tick data. For example, Hawkes processes model the irregular time spacing of the observations, event clustering and autocorrelation. However, the math is very complicated, there are very few available libraries, the computational complexity is way too high even for modest datasets. Maybe, if you were a genius or had a computational stats team to implement it for you, you could make it work, but it is infeasible for a typical quant or researcher who doesn't specialize in such methods.

The classical methods (OLS, VAR, VECM) can be quite robust and you can play with data to make it fit the methods. OLS (and hence VAR estimated with OLS) is quite robust to non-normal and heteroscedastic data. Autocorrelation is exactly what VARs are meant to model. You can say that your observations are equally spaced in event time (and it works even better than "properly" working with irregular spacing in normal wall clock time).