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The Ljung-Box statistic is not significant calculated over lags one to six lag six simply because you have no particularly large autocorrelations until lags seven & eight. Plot & examine the auto-correlation function of the residuals. There's some evidence for lack of fit, so perhaps you can come up with a better model: higher autocorrelations seem ...
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Good question.
Now obviously the results are not comparable in the sense that the F-Test you test (in this case) two linear combinations. It is therefore outside of the scope of the t-Test.
I will therefore use $H0_F$ and $H0_t$ to discriminate the two hypothesis.
With that being said, an intuitive approach to understand what is going on is to look at ...
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auto.arima can select the order of differencing automatically, and make appropriate forecasts taking account of these differences. It can also include regressors via the xreg argument, and select the appropriate model order taking account of the regressors. If you include regressors, these will be differenced along with the response variable as part of the ...
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