This assumption is not necessarily an appropriate one, and we might, for example, expect the random terms to be correlated across individuals (e.g., reflecting the state of the economy).

In the Hoenigs method, M was inversely correlated with longevity across a wide variety of taxa (Hewitt and Hoenig 2005):

And while the noise produced by oscillators is correlated across frequency, the correlation is not a set of equally spaced impulses as it is with driven systems.

Error terms can be correlated across time for the same series, individual, and horizon.

They can be correlated across time and across series for the same individual and horizon, etc.

If one believes that inaccuracies in the estimates of incidence rates are correlated across diseases, the overall uncertainty would be greater.

The data is correlated across the network, claiming a "network effect" of increased security due to this correlation.

So consumption could be negatively correlated across countries.

In addition, the lengths of homologous spacer intervals are highly correlated across species (Figure 7).

The benefits of such a mechanism would depend on the degree to which macro conditions are correlated across countries.