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I have two time series, both at a high-frequency level.

My question has two parts:

  1. How do I calculate correlation in a high-frequency setting? I assume that the normal correlation theory would not suffice.
  2. I also need to estimate the lag at which the correlation between the two series is the highest. If the series $A$ has values $a_1, a_2, a_3, \dotsc$ and series $B$ is $b_1, b_2, b_3, \dotsc$ for times $t_1, t_2, t_3, \dotsc$, I need to find the lag $t$ for which the correlation between the two series $a_i, a_{i+1}, \dotsc$ and $b_{i+t}, b_{i+t+1}, \dotsc$ is the highest.

Can you please share some resources and insights?

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  1. The property of high frequency has no statistical content per se. If I have a time series $x_t$ with values $(x_1,x_2,x_3,\dotsc)$ and change the time index from years $(2001,2002,2003,\dotsc)$ to seconds $(1,2,3,\dotsc)$, that does not change anything from a statistical point of view. However, high-frequency data may tend to have certain statistical characteristics that make application of correlation problematic. You would need to specify those characteristics if you want a concrete answer.
  2. If you do not have subject-matter knowledge on what the time lag should be, cross-correlation analysis could help. That is, you could just examine a bunch of different lags and see which one yields the highest correlation. The ccf funcion in R can do that. That would not automatically give you a confidence interval for which lag has the highest correlation in population, though.
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  • $\begingroup$ Thanks a lot for your inputs Richard, really helpful. I read a little about the ccf function and the theory behind it, makes a lot of sense and is precisely the thing I was looking for. As for the first part of the question, the series is a financial market data series. It has all the regular characteristics of such a series like bid-ask bounce and quote stuffing. Would you be able to share some insights? $\endgroup$ – nimbus3000 Apr 24 '16 at 19:43
  • $\begingroup$ Financial market data series is not precise enough. Are these prices (which are highly persistent, rendering correlation analysis largely unsuitable), returns (you would see the bid-ask bounce as first-order negative autocorrelation), volume (there will be seasonal patterns and some long-run trend, hence persistency, hence correlation analysis may be problematic), or yet something else? $\endgroup$ – Richard Hardy Apr 30 '16 at 16:25
  • $\begingroup$ Great point Richard. I'll have to go back to the data and find some of the answers to your question. Thanks a lot once again. $\endgroup$ – nimbus3000 May 1 '16 at 10:46
  • $\begingroup$ I'm not allowed to upvote it yet...not enough reputation points and dont want to mark it close yet. :-) $\endgroup$ – nimbus3000 May 2 '16 at 13:37
  • $\begingroup$ That's fine, I get it. $\endgroup$ – Richard Hardy May 2 '16 at 13:39

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