We know that Ljung-Box test can be used to test for the residuals of a fitted model. But to test for the serial correlation of a time series itself, is there a way to do that?
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1$\begingroup$ You could always just run autocorrelations on the raw response data, instead of the residuals. Have you searched the site? This strikes me as a question that's been addressed before. $\endgroup$– gung - Reinstate MonicaCommented Jan 13, 2013 at 3:51
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$\begingroup$ Yes I have searched the site but didn't get any clear answers. There should be some test for time series itself but failed to find it. $\endgroup$– yanfei kangCommented Jan 13, 2013 at 23:20
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You can still use the Ljung-Box test for the time series itself as @gung says. For a series $\{Y_t\}$ think of it as testing the residuals from the model $Y_t=0+\varepsilon_t$ if you like.
answered Jan 14, 2013 at 15:22
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$\begingroup$ @Scotchi ... I think you mean Y(t)=constant + e(t) $\endgroup$ Commented Jan 14, 2013 at 15:56
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$\begingroup$ @IrishStat Why? The sample autocorrelations that go into Ljung-Box will be the same if it's zero or a constant. $\endgroup$– Scortchi ♦Commented Jan 14, 2013 at 16:06
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$\begingroup$ You indeed might be right. Can you validate that as I don't have routine access. Post the results and copy them to me at [email protected]. Thanks in advance. $\endgroup$ Commented Jan 15, 2013 at 12:41
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1$\begingroup$ The sample autocorrelation for $k$ lags is $\sum(Y_t-\bar{Y})(Y_{t-k}-\bar{Y})/\sum(Y_t-\bar{Y})^2$, so adding a constant $c$ cancels out: $(Y_t+c-\sum(Y_t+c)/n)=Y_t+c-\bar{Y}-c=Y_t-\bar{Y}$. In any case my point was just that if you were to choose that $c=0$ model your residuals would be equal to the raw response data - illustrating why you can use the Ljung-Box test on the time series itself. $\endgroup$– Scortchi ♦Commented Jan 15, 2013 at 14:06