# Confidence intervals for autocorrelation function

Given a time series data sample I have computed autocorrelation coefficients for various lags, the result looks something like this How do I compute the confidence intervals around the sample autocorrelation curve?

The reason for that is to see if another autocorrelation curve computed from samples generated by some model is within those confidence intervals.

• Maybe try with a bootstrap method. – user2974951 Sep 24 '18 at 7:52
• @user2974951 thanks for your suggestion, I will try that. However, I was hoping to find out if some asymptotic approximation formula was available. – Anya Sep 24 '18 at 8:25

A quick google search with "confidence intervals for acfs" yielded

In there, on page 38, the standard error of an AC estimator at lag k is stated to be

$$AC_{SE,k} = \sqrt{N^{-1}\left(1+2\sum_{i=1}^k[AC_i^2] \right)}$$

where $$AC_i$$ is the AC esimate at lag i and N is the number of time steps in your sample. This is assuming that the true underlying process is actually MA. Assuming asympotic normality of the AC estimator, you can calculate the confidence intervals at each lag then as

$$CI_{AC_{k}} = [AC_{k} - 1.96\times\dfrac{AC_{SE,k}}{\sqrt{N}}, AC_{k} + 1.96\times\dfrac{AC_{SE,k}}{\sqrt{N}}]$$.