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Autocorrelation is defined as $\rho(\tau)=E((x_i-\mu)(x_{i+\tau}-\mu))/\sigma^2$ which should have a value between $\pm1$.

However, what if both $x_i$ and $x_{i+\tau}$ are significantly larger than $\mu$? Would it not be possible to have correlation greater than one? I'm particularly thinking about large lags where you only have a few sets of points before you run out of data. I'm sure this shouldn't be the case, but I can't see why.

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You're conflating sample and population calculations. The $E$ there means something. If you are talking about sample calculations, consider both numerator and denominator with care. –  Glen_b May 15 at 19:26

2 Answers 2

up vote 8 down vote accepted

I think you are missing the difference between the theoretical concept of correlation, and the empirical values you get by estimation: estimated correlations. So your question seems to be whether the empirically estimated autocorrelation can be greater than unity.

To answer that, first you need to specify which estimator of correlation you are using. In the time-series context, there are usually two possibilities: divide by $n$, the same value for each lag. That cannot give an unbiased estimator, but are usually choosen because it assures the estimated autocorrelation function is positive-definite (which is a property of the theoretical autocorrelation function we try to estimate). Or you can take into account the actual number of pairs available at the lag, which reduces by one each time lag increases by one.

That cannot guarantee the estimated autocorrelation function is positive definite, so is rarely used. But in either case, that the size of the estimated autocorrelation is $\le 1$ can be proved by the Cauchy-Schwarz inequality.

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$\sigma$ would grow too.

If you think of expectation as a scalar product you may apply Cauchy-Schwarz inequality and find that correlation is bounded by 1.

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