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EdisonCruise
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Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 5th, 2021, 11:20 am

I am working on high frequency data and try to implement an Ornstein-Uhlenbeck mean reverting process to model short-term momentum \(\alpha\). 
$$ {d\alpha_t=\eta(\theta-\alpha_t)dt+\sigma dB_t} $$
The timestamp interval between each best bid-ask sample is 0 - 30 milliseconds, irregularly.  I am confused on how to sample \(\alpha\). If I define it as the difference of mid price between each timestamp, then \(\eta\) tends to be infinity( estimated based onJose Carlos Garca Franco's paper Maximum likelihood estimation of mean reverting processes). I guess the reason is that the sampling frequency of the raw data is too high. There is some cases that the mid price changes in less than 1 millisecond, but the prceision of timestamp is up to 1 millisecond.
If I resample the data at frequency 30, 50, 100, 500 milliseconds, I can get meaningful parameters, but they vary a lot. Does it mean momentum reverse at various rate at various time scale? If resampling data is ncessary, then how to choose the resampling frequency? 
 
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bearish
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 5th, 2021, 12:56 pm

This is totally not my area, but my best guess is that you have a horribly misspecified model. To test that, you should be able to easily simulate data according to your postulated process and apply your estimation process to that dataset. If your problem persists, you need a better estimator. If not, you need a better model (or a different data set).
 
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Alan
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 6th, 2021, 5:11 pm

Likely bearish is right. But, it might be interesting to see if a difference of a weighted mid-market price would improve things.
 
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EdisonCruise
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 9th, 2021, 6:53 am

Thank you all for your suggestions. I can calibrate the model to get close parameters with simulated data sampled at difference frequencies. So I think the parameter estimation should be correct. A weighted mid-market prices based on available volumes on best bid and ask seem improve the result slightly. The \(\sigma\) reduces from 120 (mean of bid and ask) to 60 (volume weighted mid price), while \(\sigma\) is usually around 1.5 to 10 in resampled cases. But \(\eta\) still tends to infinity. Actually as the resampling period decreases, both \(\sigma\) and \(\eta\) increase.

In practical trading environment, there is certain latency between my trading server and exchange  say 30 milliseconds, and I can only update my quote every 30 milliseconds. It means I have to neglect the price update from exchange in between every 30 milliseconds. So is it justified to resample data every 30 milliseconds before I estimate the Ornstein-Uhlenbeck model?
 
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Alan
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 9th, 2021, 3:51 pm

Just out of curiosity, what do you find for [$]\eta[$] at different sampling frequencies taking [$]\alpha_t[$] to be simply the weighted mid-market price -- not a difference in prices?
 
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EdisonCruise
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 10th, 2021, 2:58 am

\(\alpha_t\) still tends to be infinity even in the case of volume weighted mid-market price.
 
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EdisonCruise
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Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

August 11th, 2021, 4:51 am

I can think of one more problem. Ornstein-Uhlenbeck process has been used to study interday mean-reversion. Usually daily data is used to estimate model parameters. But can we use millisecond level or minute level data to study interday mean-reversion? If not, it seems that, for practical financial time series, mean-reversion may demonstrate multiple time scales. And one has to resample data to match the mean-reversion time scales he wants to study.