The typical Portmanteau tests (Ljung–Box test, Box–Pierce test) actually test independence (which obviously implies autocorrelation) between successive time series observations.

I want to explicitly allow dependence but want to rule out autocorrelation up to some lag, i.e. test the null hypothesis $$ H_0 : \rho_1 = \rho_2 = \ldots = \rho_l = 0, $$ that is, "the autocorrelations up to a certain lag are all equal to zero".

Do you know of a test like this?


The Ljung-Box and the Box-Pierce tests actually do test the null hypothesis $H_0\colon \rho_1 = \rho_2 = \ldots = \rho_l = 0$ that you are interested in. They do not rule out other types of dependence, say, autoregressive conditional heteroskedasticity (autocorrelation in second moments), though the tests' null distributions are derived under conditional homoskedasticity. (Testing for independence on the other hand is practically impossible, as the variety of possible forms of dependence is infinite.)

  • $\begingroup$ Are you sure? On the English Wikipedia page of the Ljung-Box test, the H0 is independece. Also in the "Analysis of financial time series" book by Tsay it says (p.27): "Under the assumption that {r_t} is an iid sequence [...] the test stat is asymptotically chi-squared. $\endgroup$ – stollenm Sep 24 '20 at 20:40
  • $\begingroup$ @stollenm, $H_0$ is actually the one you wrote and I repeated, I am pretty positive about that. What additional assumptions there are in the model used for deriving the null distribution of the test statistic is another question. Assuming independence would be the simplest possibility. Relaxing this is possible to a degree, and the test may be more robust to some deviations from independence than other. E.g. I think the test would work quite fine under ARCH type of dependence. $\endgroup$ – Richard Hardy Sep 24 '20 at 20:51
  • $\begingroup$ ok thank you very much. I always thought that the assumptions you make while deriving an (asymptotic) distribuition of a test-stat, are implicitly what you actually test for. I will try to wrap my head around that a little more. $\endgroup$ – stollenm Sep 24 '20 at 20:56
  • $\begingroup$ @stollenm, this is not incorrect; the key term is implicitly. This holds for all statistical tests. They are based on a bunch of assumptions one of which is called $H_0$ and treated differently from the rest. The test statistic is constructed in a way that deviations from $H_0$ could be captured, but not necessarily other deviations. $\endgroup$ – Richard Hardy Sep 25 '20 at 8:01
  • $\begingroup$ @stollenm, let me know if you need some further clarification. Otherwise, consider accepting my answer (this is just a reminder; no pressure at all). $\endgroup$ – Richard Hardy Sep 29 '20 at 16:27

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