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We want to apply extreme value theory to the maximum yearly temperatures. Before we choose the model we want to test whether or not we can assume independence. First we made a plot of the autocorrelation function: acf plot. Thereafter, we applied the Ljung-Box and/or Box-Pierce test. However we find that if we only include one lag, there is no statistical evidence of dependence on the 5% level (our p-value is about 0.1) but if we include 2 or more lags, the p-values are significantly lower than our significance level. Can anyone help us understand what we can conclude from these results?

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It is difficult to answer your question without actually looking into the data not just reflecting on summary statistics like the acf/pacf. To correctly use and understand a statistical test one has to know the assumptions underlying the test ( see A. Wald https://medium.com/@penguinpress/an-excerpt-from-how-not-to-be-wrong-by-jordan-ellenberg-664e708cfc3d !) . The two tests that you mentioned both requite that the process under evaluation does not have any Pulses, Level/step shifts , seasonal pulses and.or local time trends. Furthermore the process needs to have a constant error process over time.

If you post one of your examples I will try and help. If your data is confidential then I might be able to help offline.

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