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What methods exist to test for the existence of any sort of dependence in a time series? This is in contrast to something like auto-correlation, which tests for a particular type of dependency. Is there something that can say if dependencies exist, in general, without necessarily determining what kind?

I have heard of and experimented with something called a "differential spectrum test", which states that the histogrammed changes in the time series should be symmetric about zero if the series of changes is independent. (i.e. histogram the set of $x_{t+1} - x_{t}$ and it should be symmetric about zero). If anyone has a reference for this test, I would appreciate it.

Update:

The comments suggest that failing a white noise test would indicate that some sort of dependencies exist. True? I guess it must depend on the test. Here are some I have found.. any comments on their applicability here?

Bartlett and Q Tests

Welch's Method

Method of Perez et. al.

Method of Lobato and Velasco

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    $\begingroup$ What kind of "dependence" are you looking for specifically? After all, we could take a series $(x_1,\ldots,x_n)$ that satisfies your "differential spectrum test" and set $x_{n+j}=x_j+x_n-x_1$, $j=1,\ldots,n$. This series of $2n$ values would satisfy the test--the histogram is almost the same, with a slightly higher peak at $0$--but contains a very strong dependence indeed (half the values are completely determined by the other half)! Note, too, that your test is not a test of independence: it is merely a test of symmetry of first differences, which is much weaker than independence. $\endgroup$
    – whuber ♦
    Commented Nov 16, 2011 at 20:33
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    $\begingroup$ @whuber Ideally, the test would conclude that the time series is not independent. In other words, a completely general test that could tell us when some sort of dependency exists, but not necessarily the kind of dependence. $\endgroup$
    – Pete
    Commented Nov 17, 2011 at 5:06
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    $\begingroup$ So why are you actually not looking for the test on independence, with the alternative that some kind of dependence exists? Any (weak just zero mean, constant variance, uncorrelated OR strong + identically distributed) white noise tests are welcome here. $\endgroup$
    – Dmitrij Celov
    Commented Nov 17, 2011 at 14:59
  • $\begingroup$ @DmitrijCelov Yes, failing a (strong) test for independence would fit the bill. Can you suggest anything? Is Welch's Method applicable? $\endgroup$
    – Pete
    Commented Nov 17, 2011 at 16:29

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