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I am confused with the “autocovariance” in time series. In textbooks it says that it: “measures the linear dependence between two points on the same series observed at different times”.

Why “two points”? When I see the definition of covariance it is: “measure of the tendency of two random variables to vary in the same direction“.

When I use covariance I do not do a calculation for "two points" I do it for all the points: enter image description here

And also I do not understand the logic… I do know cov(X,X)=var(X) … And in the time series I am looking I work with only one random variable? Isn’t the “autocovariance” just the “variance”?

In stats I use covariance to know what trend X vs Y has… Positive, negative or no trend. What information the “autocovariance” gives me?

If I check the correlation on a random variable against the same random variable, it is gonna be 1, what is the benefit of calculating it on a single variable in time series? isn't it always 1?

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  • $\begingroup$ Hi: the auto-covariance ( or autocorrelation ) is just the covariance ( or correlation ) of a series on the lagged version of itself. So, you take a series, $X_t$ say and the covariance at lag 1 is, the covariance of $X_t$ with itself but lagged by 1 period. So, it takes the pairs $(X_t, X_{t-1}), (X_{t-1}, X_{t-2}), \ldots (X_{t-n}, X_{t-(n+1)})$ and calculates the covariance. A similar process for lags $2, .\ldots$ and lag $n$. $\endgroup$
    – mlofton
    Commented Sep 8, 2019 at 14:04
  • $\begingroup$ Note that the covariance at lag zero is defined as the variance of the series. $\endgroup$
    – mlofton
    Commented Sep 8, 2019 at 14:07

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