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My professor gave me the sequence ${X_n} = {1,5,5,1,5,5,...}$ and asked us to compute the empirical autocovariance function given below. $$\displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} X_n X_{n+1} - \left(\frac{1}{N} \sum_{n=0}^{N-1} X_n\right) \left(\frac{1}{N} \sum_{n=1}^N X_n \right)\\$$Although I easily found this to be $\frac{-16}{9}$, this computation was the extent of the question. I would like to know the significance of this number - what does it tell me?

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It's a measure of (linear) dependence between now and one period back, albeit that the scale of $X$ matters. So it's better to look at the autocorrelation instead.

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