ACF significance band at n = 400

Question

You have n = 400 observations and plot the sample autocorrelation function. Give the approximate 95% significance band \pm 1.96/\sqrt{n}.

Accepted Answer

Under the white-noise null, each sample autocorrelation is roughly normal with standard deviation 1/\sqrt{n}, so the band is \pm \frac{1.96}{\sqrt{400}} = \pm \frac{1.96}{20} = \pm 0.098. With four times as many observations as an n=100 sample, the band shrinks by a factor of \sqrt{4} = 2 (from about 0.196 down to 0.098). More data makes the test more sensitive, so smaller autocorrelations become detectable. Takeaway. The ACF band \pm 1.96/\sqrt{n} tightens as the sample grows; quadrupling n halves the band, letting you catch weaker serial correlation.

ACF significance band at n = 400

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