Is this return series white noise?

Question

You compute the sample ACF of n = 400 daily returns. The lag-1 value is \hat{\rho}1 = 0.05, and lags 1 through 5 are all about 0.05. Is there evidence of serial correlation? Test the single lag, then test the first five jointly.

Accepted Answer

Single lag. Under the null of white noise, \hat{ho}k \approx \mathcal{N}(0, 1/n), so the 5\% significance band is \pm \frac{1.96}{\sqrt{n}} = \pm \frac{1.96}{\sqrt{400}} = \pm \frac{1.96}{20} = \pm 0.098. Since \hat{ho}1 = 0.05 < 0.098, the lag-1 autocorrelation is not individually significant. A single lag near 0.05 is well inside the noise. Joint test. Testing five lags at once, use the Ljung-Box statistic Q = n(n+2)\sum{k=1}^{h}\frac{\hat{ho}k^2}{n-k}. With h = 5, each \hat{ho}k^2 = 0.0025 and each n - k \approx 400, so Q \approx 400 \cdot 402 \cdot 0.0025 \cdot \left(	frac{1}{399…

Is this return series white noise?

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