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I've plotted 30-year moving averages across time for a couple of portfolios, and I was wondering how to calculate a 95% CI for the these moving average data (i.e., across all moving average data points, what is the CI?).

Given the points are clearly not independent of one another, I would think just using ±1.96 SD would be incorrect (as this nonindependence would be a violation of the central limit theorem), but I could be wrong and am not sure how this is done. Any links, book references, and recommended R packages/functions are also appreciated. Thank you!

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If you calculated MA by hand without actually fitting a MA(q) time series model, then you are out of luck. I suggest you use R, like in this example that shows how to construct a prediction interval, among other things.

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Maybe this paper is helpfull http://www.wiley.com/legacy/wileychi/marketmodels/chapter5.pdf

in page 8 we have Confidence Intervals for Moving Averages

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