White-noise significance band for the sample ACF

Question

You have n = 100 observations and you plot the sample autocorrelation function. Give the approximate 95% significance band \pm 1.96/\sqrt{n}, and say whether an observed \hat{\rho} = 0.15 is significant.

Accepted Answer

Under the null hypothesis that the series is white noise, each sample autocorrelation is approximately normal with mean 0 and standard deviation 1/\sqrt{n}. The 95% band is therefore \pm \frac{1.96}{\sqrt{n}} = \pm \frac{1.96}{\sqrt{100}} = \pm \frac{1.96}{10} = \pm 0.196. Any sample autocorrelation lying inside this band is statistically indistinguishable from zero. Here \hat{\rho} = 0.15, and since 0.15 < 0.196, it falls inside the band, so it is not significant: at 100 observations that value is consistent with pure noise. Takeaway. The rule-of-thumb ACF band is \pm 1.96/\sqrt{n}; a sample…

White-noise significance band for the sample ACF

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