Is this sample autocorrelation significant?

Question

From n = 256 observations, the lag-1 sample autocorrelation is \hat{\rho}1 = 0.20. Is it significant at the 5% level? Answer yes or no.

Accepted Answer

Build the white-noise significance band. Since \sqrt{256} = 16, \pm \frac{1.96}{\sqrt{256}} = \pm \frac{1.96}{16} \approx \pm 0.1225. The observed value \hat{\rho}1 = 0.20 lies outside this band (0.20 0.1225), so it is statistically distinguishable from zero. The answer is yes, it is significant: the series shows real lag-1 autocorrelation, not just noise. Takeaway. Judge a sample autocorrelation by whether it clears \pm 1.96/\sqrt{n}; at n = 256 the band is about \pm 0.12, so \hat{\rho}1 = 0.20 is a significant signal.

Is this sample autocorrelation significant?

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