Let be a stochastic process with mean , and variance . Let be an observed time series of .

A good estimator for is . We know that if the observations are IID, . However, if the observations are not IID, the variance will be larger – this is what I learned recently in Loucks (2005). The authors skipped some details of the book, so I worked them out. Below is a detailed proof.

Now, on one hand, the summands where can be grouped; on the other hand, note that when , there is one summand for and one summand for . Therefore,

(1)

Let , in other words, denotes the lag between the and timesteps. (1) becomes

where is the lag- autocorrelation and is defined as

Observe that compared to the IID case, the variance of the sample mean estimator is inflated by a factor bigger than 1. Furthermore, it can be checked that this factor does not decrease as increases. We conclude that the sample mean of an autocorrelated time series always has a bigger standard error than that of an IID time series with the same variance.

**References**

Loucks, D.P et al. (2005). *Water Resources Systems Planning and Management* (Chapter 7, p. 198-201). UNESCO Publication. (The book is publicly available on the UNESCO website)