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I have daily time series for returns on currency exchange rate. I want to estimate the frequency of crossing certain level for cumulative return, that is, how often cumulative return over several days exceeds some level.

I have checked for autocorrelation and found that coefficients are insignificant. Can I use simple bootstrap in this case or do I need to use block bootstrap or sort of? I know that correlation is not a measure of dependece and it simply shows linear dependence.

Also, should I choose a number of drawings from sample for bootstrap equal to a size of sample or it can be less?

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  • $\begingroup$ In time series ordering is important then you cannot use simple bootstrap in this case. $\endgroup$ – TPArrow Sep 28 '15 at 9:15
  • $\begingroup$ What is the criterion of this importance? $\endgroup$ – Andrew Sep 28 '15 at 9:17
  • $\begingroup$ I fail to understand what you want to do: What evaluation are you running? What does the autocorrelition have to do with how often the time series summed over some days (or moving average) exceeds some level? $\endgroup$ – Wrzlprmft Sep 28 '15 at 9:30
  • $\begingroup$ I want to estimate a number of days passed since the beginning of sample when cumulative return passes threshold $\endgroup$ – Andrew Sep 28 '15 at 9:31
  • $\begingroup$ Yes, but that does neither require bootstrapping nor the autocorrelation. You can directly measure it. $\endgroup$ – Wrzlprmft Sep 28 '15 at 9:43
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If your time series has no autocorrelation, this indicates that the time information is irrelevant. To be on the safe side, you can confirm this using other methods such as the fourier transform or by randomising your time series temporally and look whether the autocorrelation does not significantly change.

Assuming that this is correct, i.e., that the temporal information is irrelevant, you can treat your time series information as independent samples from some distribution $Ω$. Thus your problem of finding how frequently the sum of $n$ consecutive values of your time series exceeds a certain value $ϑ$ is equivalent to the problem of finding how often the sum of $n$ random samples from $Ω$ exceeds $ϑ$.

Therefore you can boostrap your findings by randomising your time series order, which is equivalent to taking another sample from $Ω$.

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  • $\begingroup$ What about bootstrap sample size? $\endgroup$ – Andrew Sep 28 '15 at 19:30

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