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What is the best way to convince myself that these data are correlated? that no univariate discrete distribution would approximate them well? that a time series model is necessary to better estimate the future distribution of counts?

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Correlated with what? Each obs to the one before it? – JoFrhwld Aug 4 '10 at 22:26
I'd hope not - the correlation at lag one is very close to 0. – rpierce Aug 5 '10 at 3:37
Or in Prof Wecker's words: "The result was a = .01, which implies little or no serial dependence in the data." – Hibernating Aug 5 '10 at 4:55
up vote 5 down vote accepted

You could just plot the ACF and check if the first coefficient is inside the critical values. The critical values are ok for non-Gaussian time series (at least asymptotically).

Alternatively, fit a simple count time series model such as the INAR(1) and see if the coefficient is significantly different from zero.

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Will the usual dw test be appropriate? – Hibernating Aug 5 '10 at 0:33
The dw test is designed for checking autocorrelation in the residuals from a regression. There is no regression here, so it is not appropriate. – Rob Hyndman Aug 5 '10 at 0:57
I agree. Books mislead again. – Hibernating Aug 5 '10 at 4:47

Immediately, a runs test comes to mind. That is, if this is an ordered set and you are interested in knowing if the ordered observations are independent. It tests whether the sequence is likely due to randomness rather than systematic variation. Without more detail, that's the best I can come up with.

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