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I've got the summary of a data set, but not the actual data. They've calculated confidence intervals using bootstrapping. I know the sample size, so if they were normally calculated confidence intervals, I'd know how to calculate standard deviation, but is there any relationship between the size of confidence intervals calculated from bootstrapping and sd?

Also, can it be said that doubling sample size sees the width of confidence interval reduce 25%/ when they have been calculated with bootstrapping?

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"is there any relationship between the size of confidence intervals calculated from bootstrapping and sd?"

Yes if the model residuals are perfectly normally distributed they would be identical ... non-normal residuals would lead to a mismatch.

I don't think that increasing the number of monte-carlo samples would reduce anything ,,,just get a better handle on the probability distribution.

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  • $\begingroup$ The pairs bootstrap technique when used (instead of residuals bootstrap) retains the dependence structure between $\epsilon$ and $X$ so that residuals and estimates even from a non-normal distribution do not have to be corrected with heteroskedasticity-consistent (HAC) standard errors (Freedman 1981, Davison and Hinkley, 1997) $\endgroup$ – develarist Nov 29 '19 at 17:57
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    $\begingroup$ Also the original question has to be read carefully. It's obvious that standard errors can be backed out of an estimated confidence interval, but the question is if standard errors can be backed out of a bootstrapped confidence interval, which usually is obtained from running $B=200$ replications on $B=200$ resampled versions of the original data $\endgroup$ – develarist Nov 29 '19 at 18:06
  • $\begingroup$ In my (time series) world the dependence structure between e and X is nul or nearly nul as a result of identifying a good model structure vitiating/ameliorating such dependence. $\endgroup$ – IrishStat Nov 29 '19 at 18:10
  • $\begingroup$ Thanks for understanding @develarist indeed I know exactly how I'd calculate standard deviation from confidence intervals calculated by t*sd/SQRT(n) but I'm working with confidence intervals calculated with the residuals bootstrap technique. I'm interested in whether increasing the original sample size (rather than the number of Monte-carlo samples) reduces the width of the bootstrapped confidence interval according to any straightforward relationship. $\endgroup$ – albecq91 Dec 3 '19 at 11:25
  • $\begingroup$ You might consider generating an ARIMA (0,0,0) process for say 1000 values. Use the first 100 and evaluate the bootstrap .. then use the first 200 .. then use the first 300 etc ... then examine for differences $\endgroup$ – IrishStat Dec 3 '19 at 11:43

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